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Utah Water Research Laboratory
1-1-1965
Design, Calibration, and Evaluation of a
Trapezoidal Measuring Flume by Model Study
M. Leon Hyatt
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Hyatt, M. Leon, "Design, Calibration, and Evaluation of a Trapezoidal Measuring Flume by Model Study" (1965). Reports. Paper 386.
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DESIGN, CALIBRATION, AND EVALUATION
O F A TRAPEZOIDAL MEASURING F L U M E
BY MODEL STUDY
bY
Milton L e o n Hyatt
A thesis submitted in partial fulfillment
of t h e r e q u i r e m e n t s f o r the d e g r e e of
of
MASTER O F SCIENCE
in
Civil Engineering
Approved:
UTAH S T A T E UNIVERSITY
L o g a n , Utah
ACKNOWLEDGEMENTS
T h e r e i s no l i m i t to the thanks and a p p r e c a t i o n that m u s t b e
given to Gaylord Skogerboe f o r the guidance, aid, and s o u r c e of i d e a s
which have been invaluable throughout this investigation.
Gratitude
i s e x p r e s s e d to D r . Calvin Clyde f o r the counsel and i n s t r u c t i o n
r e c e i v e d f r o m him.
A l s o , the suggestions offered by D r . B r u c e
A n d e r s o n , the e d i t o r a l a s s i s t a n c e of M r s . C. W. L a u r i t z e n , and the
typing of the t h e s i s by M i s s B a r b a r a South a r e v e r y m u c h appreciated.
Thanks a r e a l s o given t o t h e Utah W a t e r R e s e a r c h L a b o r a t o r y and its
staff f o r s e r v i c e s r e n d e r e d .
Appreciation would not be c o m p l e t e with-
out recognizing the help, devotion, and love given by m y wonderful
wife, J e r r o l y n .
Milton Leon Hyatt
T A B L F O F CONTENTS
Page
.
.
, . . . .
1
. . . .
4
DESIGN O F P R O T O T Y P E T R A P E Z O I D A L MEASURING F L U M E
1I
.
16
S C O P E A N D P U R P O S E O F INVSSTIGATION
L I T E R A T U R E REVIEW
. . . . . . .
,
DESIGN O F M O D E L T R A P E Z Q I D A L MEASURING F L U M E
.
. . . . . . . . .
CHANGES RESULTING F R O M MODEL STUDY . . .
EXPERIMENTAL FACILITIES
22
DESIGN
27
.
ANALYSIS O F DATA F R O M F I N A L DESIGN
S u p e r - c r i t i c a l flow a n a l y s i s
S u b m e r g e d flow a n a l y s i s
.
.
.
.
.
.
.
39
.
43
.
. . . . . . . . . .
59
65
. . . . . . . . . . .
68
. . . . . . . . . . . . .
70
S E L E C T G D BIBLIOGRAPHY
APPENDIXES
.
.
ENERGY LOSSES IN MEASURING F L U M E S
SUMMARY AND CONCLUSIONS
39
Appendlx A . D a t a and C o m p u t a t i o n s
. . . . . . .
Appendix B. S u b m e r g e d F l o w C o m p u t e r P r o g r a m a n d
Data
. . . . . . . . . . . . . .
iii
71
80
LIST O F FIGURES
Page
-
Figure
1
Canal "B" tailwater--discharge c u r v e a t s i t e of
. ,
proposed t r a p e z o i d a l m e a s u r i n g flume
. .
.
12
2
Initial prototype trapezoidal m e a s u r i n g flume design
.
14
3
Initial model trapezoidal m e a s u r i n g flume design
. .
20
Constructed m o d e l trapezoidal m e a s u r i n g flume
composed of sections 1 through 5
.
.
. . .
23
Model flume placed i n 5-foot by 5-foot flume
. . .
23
. .
23
.
.
Side view of Skction 3 showing stilling wells
,
Schematic view of laboratory facility
Top view of flume looking u p s t r e a m
.
.
. .
. . . .
. . . .
24
25
U p s t r e a m view of flume showing end of copper tubing
and point g a g e .
.
.
.
. .
.
25
Tailgate used to adjust tailwater depth
.
25
.
30
.
.
. .
. .
. . .
Downstream view of f l u m e showing s e p a r a t i o n with
.
. . . . .
flow adhering to l e f t s i d e
.
.
Top view of flume looking downstream showing
s e p a r a t i o n with flow adhering t o right s i d e
. . .
Top view of flume exit a t design flow with s e v e r a l
vanes . . .
.
.
. .
. . . . . .
30
. .
30
Top view of flume with t h r e e vanes a t end of t h r o a t
.
30
Downstream view of f l u m e a t design flow (2.3 1 c f s )
.
.
,
.
with one row of t h r e e blocks
.
31
. .
31
.
.
.
Downstream view of flume a t design flow with pne
. .
.
row of four blocks
. .
.
.
LIST OF FIGURES (continued)
Page
-
Figure
17
Design flow with two rows of blocks placqd a t the
beginning of the exit
.
. . . . .
.
. .
Design flow with two rows of blocks p l a c e d i n the
m i d d l e of the exit
. . .
. . . . . . .
U p s t r e a m view of design flow conditions with Section
I removed
. . . . . . .
D o w n s t r e a m view of 6 : 1 exit
. . . . . . .
Top view of f l u m e with 6 : 1 d i v e r g i n g exit
. . .
. . . . . .
D o w n s t r e a m view of f l u m e a t d e s i g n flow with one
vane
,
,
-
. .
.
.
.
. .
.
D o w n s t r e a m view of f l u m e a t d e s i g n flow using two
,
v a n e s and one column
. . .
.
. . . .
D o w n s t r e a m view of f l u m e a t d e s i g n flow using two
columns and one vane
,
. . .
. . . . .
. . . .
D o w n s t r e a m view of final model d e s i g n
U p s t r e a m view of f i n a l model d e s i g n
.
~
U p s t r e a m view of flume at d e s i g n conditions
. . .
. . .
D o w n s t r e a m view of f l u m e flowing a t 2. 31 cfs,fully
submerged
.
,
.
a
a
- .
. . . . . .
D o w n s t r e a m view of flume with flow of 1 . 4 c i s
. .
D o w n s t r e a m view of f l u m e d i s c h a r g i n g 1 . 4 c f s , fully
. , .
submerged
.
.
. . . . . . . .
F i n a l d e s i g n of m o d e l t r a p e z o i d a l m e a s u r i n g flume
F i n a l d e s i g n of m o d e l prototype m e a s u r i n g f l u m e
.
.
31
LIST O F FIGURES (continued)
Figure
33
34
35
36
Page
Development of r e l a t i o n s h i p between d i s c h a r g e
and u p s t r e a m depth
41
Calibration curve for critical-depth trapezoidal
m e a s u r i n g flume
42
. . . . . . . . . .
. . . . . . . . . . . .
.
E f f e c t of s u b m e r g e n c e on the d i s c h a r g e relationship
45
Relationship between e n e r g y l o s s p a r a m e t e r and log
of s u b m e r g e n c e
.
48
. . . . . . . .
49
.
.
.
.
.
.
.
.
37
Relationship between p i - t e r m s
38
G r a p h i c a l solution of m i n i m u m depth in t h r o a t
39
Relationship between F r o u d e n u m b e r , m i n i m u m
depth in t h r o a t , and d i s c h a r g e
40
41
42
.
.
.
51
.
Development of r e l a t i o n s h i p b e t w e e n d i s c h a r g e ,
F r o u d e n u m b e r , and m i n i m u m depth in t h r o a t
.
.
54
Development of r e l a t i o n s h i p between d i s c h a r g e ,
.
e n e r g y l o s s , and s u b m e r g e n c e .
Calibration curves for submerged trapezoidal
m e a s u r i n g flume
.
.
.
.
.
.
43
Design of r e c t a n g u l a r m e a s u r i n g f l u m e
44
Design of P a r s h a l l m e a s u r i n g flume
.
52
.
.
56
.
57
.
. . . . . .
61
62
NOMENCLATURE
Symb 01
A
Definition
A r e a , it.
L
A r e a a t e n t r a n c e of f l u m e , ft.
A r e a a t t h r o a t of f l u m e , f t .
2
2
A r e a a t rectangular flume entrance, it.
A r e a a t r e c t a n g u l a r f l u m e t h r o a t , ft.
A r e a i n m o d e l , ft.
M i n i m u m a r e a , ft.
2
2
2
2
A r e a in prototype, ft.
2
R a t i o of A /A , d i m e n s i o n l e s s
P
m
B o t t o m w i d t h , ft.
C o n s t a n t e q u a l t o 0 . 9 5 - 1. 0 but t a k e n , c o n s e r v a t i v e l y ,
a s 0.95, d i m e n s i o n l e s s
Coefficient u s e d t o obtain a c t u a l d i s c h a r g e , d i m e n s i o n l e s s
2
C o e f f i c i e n t d e f i n e d b y C / ~ ( A ~ /) A- 1, d i m e n s i o n l e s s
2
Coefficient defined b y h
/(Fmax)', f t .
m
-
Any d e p t h , f t .
Minimum specific energy, ft
Goefflcient of s u r f a c e d r a g , a f u n c t l o n of R e y n o l d s n u m b e r
and r e l a t i v e r o u g h n e s s , d i m e n s i o n l e s s
F r o u d e n u m b e r , dimensionless
G r a v ~ t yf o r c e s , l b s
Vlll
NOMENCLATURE ( c o n t i n u e d )
Symbol
F~
F
F
max
r
Definition
Inertia force, lbs.
Maximum Froude number in the flume, dimensionless
R a t i o of p r o t o t y p e F r o u d e n u m b e r t o t h a t in m o d e l ,
dimensionless
A c c e l e r a t i o n d u e t o g r a v i t y , 3 2 . 2 ft./sec.
2
D i f f e r e n c e i n w a t e r l e v e l s a t t h e e n t r a n c e and t h r o a t of
r e c t a n g u l a r f l u m e , ft.
U p p e r head of P a r s h a l l f l u m e , ft.
Head l o s s , i t .
Depth of flow at e n t r a n c e , f t .
Depth of flow a t point in t h e t h r o a t , f t .
Depth of flow at e x i t , f t .
M i n i m u m d e p t h of flow in t h r o a t , f t .
Depth of flow a t e n t r a n c e of p r o t o t y p e , f t .
M i n i m u m d e p t h of flow in t h r o a t f o r p r o t o t y p e , f t .
Depth of flow a t exit of p r o t o t y p e , ft.
Depth of flow at e n t r a n c e of m o d e l , f t .
M i ~ n i m u md e p t h of flow i n t h r o a t f o r m o d e l , f t .
Depth of flow at e x i t of m o d e l , f t .
Any l e n g t h , i t ,
NOMENCLATURE (continued)
Symbol
Definition
L e n g t h i n m o d e l , it
Length in p r o t o t y p e , it.
R a t i o of L /L , d i m e n s i o n l e s s
D
m
Actual d i s c h a r g e , c f s
Theoretical discharge, cfs
Discharge in model, cfs
Discharge in prototype, cfs
R a t i o of Q /Q,.
P
dimensionless
Hydraulic m e a n depth, ft,
Slope, d i m e n s i o n l e s s
Water s u r f a c e width, f t .
M i n i m u m w a t e r s u r f a c e w i d t h , ft.
W a t e r s u r f a c e w i d t h of p r o t o t y p e , f t .
Average velocity, f p s
Average velocity a t entrance, fps
A v e r a g e v e l o c i t y a t point i n the t h r o a t , i p s
Velocity i n m o d e l , f p s
Velocity i n p r o t o t y p e , i p s
NOMENCLATURE (con-cxnued)
Symbol
Definition
vr
Ratio of V / V , d i m e n s i o n l e s s
P m
W
T h r o a t width of P a r s h a l l f l u m e , it.
Section f a c t o r dependent upon g e o m e t r y , ft.
2.5
E n e r g y coefficient which c o n s i d e r s non-uniform v e l o c i t y
d i s t r i b u t i o n , equals 1. 0 , d i m e n s i o n l e s s
Coefficient depending on s t r e a m tube c u r v a t u r e , a s s u m e d
equal 1. 0 , d i m e n s i o n l e s s
Density of fluid, l b s . - s e c .
Any angle, d i m e n s i o n l e s s
2
/ft.
4
SCOPE A N D P U R P O S E O F INVESTIGATION
T h e d i s c h w g e o c c u r r i n g i n a n open channel c a n b e m e a s u r e d by
p l a c i ~ ga c o n s t r i c t i o n in the channel.
c o n s t r i c t i o n s in open charmels,
F l u m e s a r e commonly u.sed a s
A flume i s a s p e c i a l l y designed and
c a l i b r a t e d s e c t i o n built into a channel, t h e p h y s i c a l p r o p e r t i e s of
which allow t h e calculation of the d i s c h a r g e .
of the f l u m e i s u s u a l l y called the t h r o a t .
T h e n a r r o w e s t section
The velocity of flow through
the t h r o a t , f o r any given flow r a t e , i n c r e a s e s with a d e c r e a s e in the flow
deptlp.
The ideal condition f o r m e a s u r e m e n t of d i s c h a r g e i s a t h r o a t
sufficiently c o n s t r i c t e d t o p r o d u c e c r i t i c a l - d e p t h in the t h r o a t .
When-
e v e r the g e o m e t r y of a channel produces c r i t i c a l flow the relationship
between d i s c h a r g e and h e a d is independent of conditions d o w n s t r e a m ,
making d i s c h a r g e a function of only the u p s t r e a m depth.
Thus, when
critical-depth o c c u r s i n the t h r o a t , the only m e a s u r e m e n t r e q u i r e d to
d e t e r m i n e t h e d i s c h a r g e through t h e f l u m e is the u p s t r e a m depth of
flow, thus m a k i n g the wide u s e of c r i t i c a l - d e p t h f l u m e s d e s i r a b l e f o r
measurement purposes.
F l u m e s of v a r i o u s s h a p e s a r e used to obtain
a condition of critical-depth, the m o s t c o m m o n and well known being
the P a r s h a l l flume.
One p u r p o s e of thrs ~ n v e s t l g a t i o nh a s b e e n to study the tsapezordal
shaped f l u m e s which s e v e r a l r e s e a r c h e r s ( A c k e r s a n d H a r r r s o n , 1965,
L u d w ~ gand Ludwig, 1951, P a l m e r and Bowlus. 1936; Robinson and
C h a m b e r l a i n , i 9 6 2 ; awii Mi e l i s and Goiaas , 1943) have investigated.
However, t h e p r l r r ~ a r yp u r p o s e of t h i s investigation h a s been the desi-gn,
c a l i b r a t i o n , and evaluation, by m o d e l study, of a t r a p e z o i d a l m e a s u r i n g
flume to be c o n s t r u c t e d i n the distribution s y s t e m of the D. M. A . D.
Company ( D e l t a , M e l v i l l e , A b r a h a m , and D e s e r e t I r r i g a t i o n C o m p a n i e s )
in Delta, Utah.
The f l u m e l o b e c o n s t r u c t e d will be used to m e a s u r e
having a capacity of 3 0 0 c f s
i r r i g a t i o n w a t e r s in a c a r ~ a l(Canal "B")
(cubic f e e t p e r second) and located below the D. M . A . D. Dam.
The e s s e n t i a l objectives of the model study have been:
(1)
investigation of s e v e r a l e n t r a n c e and exit conditions to obtain the m o s t
economical, efficient, and p r a c t i c a l d e s i g n , (2) c o r r e l a t i o n of the d a t a
f r o m this study with that of previous r e s e a r c h , and ( 3 ) c o m p a r i s o n of
head l o s s e s i n t r a p e z o i d a l f l u m e s with t h o s e of r e c t a n g u l a r and
P a r shall flumes.
The t r a p e z o i d a l f l u r r ~ eh a s b e e n designed a s a c r i t i c a l - d e p t h
flume utilizing p r e s e n t t a i l w a t e r coriditions ( t h e p r e s e n t depth-.discharge
reltitionship f o r Canal
"B" i s i l l u s t r a t e d in F i g u r e l ) , However, in-
c r e a s e d developments by the D. M. A. D. Company i n the channel downs t r e a r n f r o m t h e proposed flume m a y yield i a c r e a s e d depth. of flow f o r
ariy p a r t i c u l a r d i s c h a r g e , thereby increasirlg the d e g r e e of s u b m e r g e n c e .
%n c a s e the t a i l w a t e r depths s h ~ > o i d
r i s e m u c h above the p r e s e n t i e v e i . ~
l o r any p a r t i c u l a r d i s c h a r g e , sut:mer6ence of the f l u m e will un..
dojii-,tedly o c c u r a n d iheii two varii;bie$ will have to b e m e a s u r e d - -
both the u p s t r e a m and t a i l w a t e r depths.
Consequently, t h e calibratlor?
of the t r a p e z o i d a l m e a s u r i n g f l u m e was extended t o s u b m e r g e d flow i n
this investigation.
After the prototype s t r u c t u r e h a s b e e n c o n s t r u c t e d , a field
c a l i b r a t i o n will b e conducted.
This field c a l i b r a t i o q will b e c o m p a r e d
with the c a l i b r a t i o n f o r the prototype s t r u c t u r e a s p r e d i c t e d f r o m t h e
model study.
Neither the field calibration n o r t h e c o m p a r i s o n between
the field and m o d e l p r e d i c t i o n calibrations will b e i n c o r p o r a t e d into
this thesis.
LITERATURE REVIEW
Although t h e r e i s a g r e a t d e a l of information a v a i l a b l e concerning
c r i t i c a l - d e p t h f l u m e s , few studies have b e e n m a d e r e g a r d i n g t r a p e zoidal f l u m e s .
The m o s t common and widely u s e d open c h a n ~ e w
l ater-
m e a s u r i n g device i s probably the P a r s h a l i m e a s u r i n g f l u m e , whlc'n is
of r e c t a n g u l a r s h a p e with a n i r r e g u l a r bottom.
The ParshaS? flume
h a s been s o designed that d i s c h a r g e m e a s u r e m e n t of f r e e and subm e r g e d flow c a n b e attained.
only of the u p s t r e a m depth.
The free-flow d i s c h a r g e i s a function
D i s c h a r g e f o r s u b m e r g e d flow through a
P a r s h a l l f l u m e i s a function of the u p s t r e a m depth and the r a t i o of the
u p s t r e a m to t h r o a t depth ( s u b m e r g e n c e ) ( P a r s h a l l , 1945, 1950, 1953).
The s u b m e r g e n c e c h a r t s provide a d i s c h a r g e c o r r e c t i o n which i s subt r a c t e d f r o m t h e f r e e - f l o w d i s c h a r g e based on t h e u p s t r e a m depth
alone.
Development of t r a p e z o i d a l flumes f o r m e a s u r e m e n t of flows in
c i r c u l a r conduits w a s accomplished by P a l m e r and Bowlus (1936).
L a t e r , Wells and Gotaas (1948) c a l i b r a t e d a n u m b e r of trapezoidal
flumes f o r u s e in c i r c u l a r conduits.
An extensive study of t r a p e z o i d a l f l u m e s f o r u s e in open channel
flow m e a s u r e m e n t h a s b e e n m a d e by Robinson and C h a m b e r l a i n (196%).
They conclyded thar a p p r o a c h condinons have only slight effects on
t h e d i s c h a r g e , and t h a t t r a p e z o i d a l flumes could o p e r a t e a t higher
d e g r e e s of sv.l,'mcrgence than rectangul.ar sections without the need
of a c o r r e c t i o n f a c t o r .
Robinsol? and Chamberlain i1962) found thz'c
trapezoidal f l u m e s could be used withoilt c o r r e c t r o n f o r sfibmergences
a s high a s 8 0 to 85 p e r c e n t .
Wells and Gotaas (1948) d e i ~ n e dsub-
m e r g e n c e a s "the p e r c e n t r a t i o of tailwater depth -lo the u-pstrearn
depth of flow w h e r e the tailwater depth i s r e f e r r e d t o channel xnvert
a t the point of u p s t r e a m m e a s u r e m e n t .
"
The justification and advantages for u s e of the trapezoidal flume
over the r e c t a n g u l a r f l q m e a s l i s t e d by Robinson and C h a m b e r l a m
(1960) a r e :
1.
Approach conditions seemed to e x e r t a m i n o r effect
on the h e a d - d i s c h a r g e relationship. M a t e r i a l d e posited in the approach sectign did not change this
relationship t o any d e g r e e .
2.
A l a r g e range of flow can be m e a s u r e d with a
relatively s m a l l change in depth thus minimizing
the amount of f r e e b o a r d needed on the c a n a l .
3.
The trapezoidal shape f l t s the common c a n a l section
m o r e c l o s e l y than does the rectangular flume.
4.
Trapezoidal f l u m e s operate under higher d e g r e e s of
submergence than will the rectangular f l u m e s without c o r r e c t i o n s bcing n e c e s s a r y t o the s t a n d a r d
rating.
The b a s l c t h e o r y lnvolved in the deszgn of the flvrne i s that
Summal-ized frron?.
minimum specific e n e r g y o c c u r s at critical-depth,
A c k e r s and H a r r i s o n (1963)
E = Bd
3
4- a: . - ~ r J 2 ~2 ,
.
.
.
.
.
.
.
.
.
,
1
i n which
E
=
minimum specific energy, f t ~
V
=
a v e r a g e v e l o c i t y of flow, f t . / s e c .
g
=
a c c e l e r a t i o n due t o gravity, 3 2 . 2 f t . / s e c .
d
=
d e p t h of f l o w , i t .
a
=
e n e r g y c o e f f i c i e n t which c o n s i d e r s n o n - u n i f o r m velocity
2
distribution, equals 1, 0, dimensionless
B
=
c o e f f i c i e n t depending on m e a n s t r e a m t u b e c u r v a t u r e ,
a s s u m e d e q u a l to 1. 0, d i m e n s i o n l e s s
F r o m t h e equation of continuity
Q=AV
.
.
~
~
.
.
. . .
i n which
A
=
c r o s s - s e c t i o n a l a r e a of f l o w , f t .
A
=
d(b
b
=
t h r o a t width, feet
1 :m =
+ m d ) f o r trapezoidal
2
shape, ft.
2
flume side slope (vertical : horizontal)
With p r o p e r s u b s t i t u t i o n a n d s o l u t i o n of t h e s p e c i f i c e n e r g y and
continuity e q u a t i o n s i t c a n b e shown that
Chow (1959) a n a l y z e d t h e flKme g e o m e t r y a c c o r d i n g t o
in which
Z
=
s e c t i o n f a c t o r dependent upon g e o m e t r y
F o r a trapezoidal shape,
Z
r e l a t e s t h e bottom width to the
depth of flow and by u s e of F i g u r e 4 - 1 in Chow (1959) a check is
provided on t h e validity of Equation 3.
4
1
m ( v a r i e s in l o c a t i o n )
I
Profile
The a p p r o a c h used by Roblnson and C h a m b e r l a i n (1962) i s the
b a s i c p r i n c i p l e of e n e r g y c o n s e r v a t i o n .
T h e B e r n o u l l i equation c a n be
w r i t t e n between two s e c t i o n s (Sections 1 and 2 ) a s
i n which
V1
V2
hl
hZ
hL
=
a v e r a g e velocity of flow a t e n t r a n c e , f t , / s e c
=
a v e r a g e velocity of flow a t a point in the t h r o a t , f t , / s e c .
=
depth of flow a t e n t r a n c e , ft.
=
depth of flow a t a point in the t h r o a t , f t .
=
l o s s of e n e r g y ( a s s u m e d negligible), f t
When combined with the equation of continuity
Q =AV
r e s u l t s in
To obtain t h e a c t u a l d i s c h a r g e , the t h e o r e t i c a l d i s c h a r g e m u s t b e
modified by a c o e f f i c i e n t , C .
Since the f i r s t r a d i c a l i s d i m e n s i o n l e s s and tends to r e f l e c t the g e o m e t r y
of the s t r u c t u r e , it c a n be i n c o r p o r a t e d into a new d i s c h a r g e coefficient,
C ' (Robinson and C h a m b e r l a i n , 1960) which yields
in which
A c k e r s and H a r r i s o n (1963) h a v e i n v e s t ~ g a t e denergy l o s s e s
through t r a p e z o i d a l f l u m e s , Head l o s s i s c o n s i d e r e d Important 1 2 t h i s
study s i n c e a m a x i m u m l o s s of one foot i s t o be allowed f o r the
prototype s t r u c t u r e under investigation.
Consequently, the r e s e a r c h
of A c k e r s and H a r r i s o n h a s b e e n utilized to p r e d i c t l o s s e s .
The t o t a l
e n e r g y l o s s through the flume c o n s i d e r s , ( a ) the effect of c o n v e r g e n c e ,
( b ) the e f f e c t of n o n - p a r a l l e l flow a t the c o n t r o l section, ( c ) f r i c t l o n
l o s s e s through the t h r o a t , and ( d ) the effect of d i v e r g e n c e .
The~w
r ork
shows the head l o s s due to f r i c t i o n through the t h r o a t to b e
in which
h
f
= head Loss, f t .
L
= length of t h r o a t , f t .
R
= hydraulic m e a n depth in t h r o a t a n d i s A/P,
f
=
ft.
d i m e n s i o n l e s s coefficient of s k i n d r a g , depending on
Reynolds n u m b e r and t h e r e l a t i v e roughness.
The head l o s s due to f r i c t i o n i s b a s e d on t h e following a s s u m p t i o n s
( A c k e r s and H a r r i s o n , 1963):
1.
Critical-depth o c c u r s throughout the length of the t h r o a t .
2.
The control s e c t i o n o c c u r s a t the d o w n s t r e a m end of t h e
throat.
3.
The velocity in the t h r o a t i s uniform.
DESIGN O F PROTOTYPE TRAPEZOIDAL
MEASURING FLUME
The approach used f o r designing the prototype t r a p e z o i d a l
m e a s u r i n g f l u m e w a s t h a t given by A c k e r s and H a r r i s o n (1963) in
Equation 3.
The d e s i g n w a s t h e n checked by the m e t h o d d e v e l o p e d
b y Chow (1959). To obtain t h e n e c e s s a r y d i m e n s i o n s , t h e r e s e a r c h b y
Robinson and C h a m b e r l a i n (1960), and A c k e r s and H a r r i s o n (1963) w a ?
consulted.
The side s l o p e of the t r a p e z o i d a l s e c t i o n h a s b e e n found ta
be "unimportant provided i t s a t i s f i e s the n e c e s s a r y e n e r g y r e l -at 'ionships"
( P a l m e r and Bowlus, 1936; Wells and G o t a a s , 1948). T h u s ,
the author decided upon a 1 : 1 s i d e slope.
The p r e s e n t o r e x i s t i n g d e p t h - d i s c h a r g e relationship in C a n a l
"B" ( a s shown i n F i g u r e 1 ) gives the tailwater depth f o r any d i s c h a r g e .
The velocity in the c a n a l i s v e r y low.
T h e r e f o r e , the minimurn specific
e n e r g y w a s taken a s t h e depth of flow in the c a n a l and the velocity head
w a s neglected.
H e n c e , b a s e d on the range of d i s c h a r g e s t o b e m e a -
s u r e d b y the f l u m e , which i s 2 0 to 300 cfs (with 300 c f s being used a s
the d e s i g n flow), the t h r o a t width w a s calculated to be six f e e t u s i n g
Equations 3.
This s a m e value of s i x f e e t w a s calculated f r o m
Equation 4 and u s e d a s a c h e c k on the validity of Equation 3 .
In Equation 1 , an a n a l y s i s of the actual exit velocity d i s t r i b u t i o n
o c c u r r i n g in t h e m o d e l showed a = I . 0, and
i,O.
B w a s a s s u m e d equal t o
Calculations b a s e d on t h i s equation show tha: f r e e - f l o w wiLI occur
over the range of d i s c h a r g e u n d e r p r e s e n t t a i l w a t e r c o n d i t i o n s ,
No definite c r i t e r i a f o r the length of the t h r o a t c a n b e found in
the l i t e r a t u r e .
the head.
A c k e r s a n d H a r r i s o n (1963) suggest a length of twice
A length of two to f o u r t i m e s the t h r o a t width i s suggested by
Robinson and C h a m b e r l a i n (1960).
The a u t h o r ' s opinion, a f t e r con-
s i d e r a t i o n of t h e i r r e s e a r c h , i s that a t h r o a t length of f i f t e e n f e e t ,
which i s two and one-half t i m e s t h e t h r o a t width, would b e sufficient.
F r o m A c k e r s and H a r r i s o n (1963) the calculated f r i c t i o n a l l o s s e s
i n the t h r o a t w e r e found to b e about one-tenth of the t o t a l l o s s .
Since
t h e f r i c t i o n a l l o s s e s through t h e t h r o a t + r e only a s m a l l p e r c e n t a g e of
the total e n e r g y l o s s , t h e a u t h o r felt that s e v e r a l e n t r a n c e and exit
conditions should be investigated.
The b a s i c c r i t e r i a f o r u s e of a
s h o r t e r o r l o n g e r e n t r a n c e o r exit t r a n s i t i o n slope depend on the l o s s
of energy, a c c u r a c y of flow m e a s u r e m e n t r e q u i r e d of t h e installation,
and possible e r o s i o n d o w n s t r e a m f r o m the flume.
Evidence shows that
the "exit t r a n s i t ~ o nh a s n o effect on the a c c u r a c y of m e a s u r e m e n t and
.
.
,
i s d e s i r a b l e only to c o n s e r v e energy" (Wells and G o t a a s , 1948).
T h i s s t a t e m e n t i s t o b e modified f o r e a r t h c h a n n e l s , w h e r e high exit
velocities m a y c a u s e e r o s i o n .
The w o r k s of s e v e r a l a u t h o r s ( A c k e r s and Harkison, 1963; Ludwig
and Ludwig, 1951; P a l m e r and Bowlus, 1936; Robinson and C h a m b e r l a i n ,
1960; Wells and G o t a a s , 1948) w e r e investigated to d e t e r m i n e the
d e s i r a b l e e n t r a n c e and exit t r a n s i t i o n slope. b c k e r s and H a r r i s o n
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DESIGN O F M O D E L TRAPEZOIDAL MEASURING F L U M E
T h e p r i m a r y o b j e c t i v e s of the m o d e l s t u d y of t h e p r o p o s e d
p r o t o t y p e f l u m e t o b e u s e d b y the D.M.A. D. C o m p a n y w e r e :
(1) the
i n v e s t i g a t i o n of e n t r a n c e and exit conditions t o f i n d , if p o s s i b l e , a
m o r e e c o n o m i c a l d e s i g n t h a n t h a t which w a s f i r s t p r o p o s e d ,
(2)
c a l i b r a t i o n a n d r e c o m m e n d a t i o n of t h e d e s i g n t o t h e D. M. A . D.
C o m p a n y , a n d ( 3 ) c o r r e l a t i o n of the d a t a f r o m t h i s s t u d y w i t h t h o s e of
previous investigations.
The t e a c h i n g l a b o r a t o r y a t Utah S t a t e U n i v e r s i t y i s equipped w i t h
a five-foot-wide f l u m e , which i s o n e - s e v e n t h t h e w i d t h of t h e 35-foot-
w i d e c a n a l into which t h e p r o t o t y p e f l u m e i s t o b e p l a c e d .
The l a b o r a t o r y
f l u m e is of r e c t a n g u l a r s h a p e , w h e r e a s C a n a l "B" i s U - s h a p e d .
s i m u l a t e t h e g e o m e t r i c s h a p e of Canal
To
"B" i n t h e l a b o r a t o r y , it would
b e n e c e s s a r y t o f i l l i n t h e c o r n e r s of t h e f l u m e .
Robinson and
C h a m b e r l a i n (1960) h a v e found t h a t a p p r o a c h c o n d i t i o n s h a v e a m i n o r
e f f e c t on t h e h e a d - d i s c h a r g e r e l a t i o n s h i p a n d a r e t h e r e f o r e not s i g n i f i c a ~ t
C o n s e q u e n t l y , t h e c o r n e r s of the f l u m e w e r e not f i l l e d .
After consider;
a t i o n of the a v a i l a b l e l a b o r a t o r y f a c i l i t i e s , a d e c i s i o n w a s m a d e t o u s e
a Length r a t i o of
1 : 7 (model : prototype).
In the open c h a n n e l flow p r o b l e m b e i n g s t u d i e d , l a m i n a r flow
w i l l not o c c u r a n d s u r f a c e t e n s i o n w i l l h a v e n o s i g n i f i c a n t e f f e c t ( A c k e r s
and H a r r i s o n ,
1963).
The p r e d o m i n a n t f o r c e s a c t i n g on t h e flow w i l l
b e t h o s e of gravi'cy a n d i n e r t i a .
T h e g r a v i t y f o r c e s a r e pronounced
when the flow i s s u b c r i t i c a l and the i n e r t i a f o r c e s a r e pronoti.nced wtien
the flow i s s u p e r c r i t i c a l .
prototype.
Both f o r c e s a r e e x e r t e d on the m o d e l and
The r a t i o of i n e r t i a f o r c e s to g r a v i t y f o r c e s i s the F r o u d e
The i n e r t i a f o r c e s a r e given by
number.
in which
FI = i n e r t i a f o r c e s , l b .
4
p
= density of fluid, 1b.-see. /ft.
L
= length, f t .
V
= a v e r a g e velocity of flow, ft. / s e c .
The g r a v i t y f o r c e s a r e given by
in which
F
g
g
= gravity f o r c e s , lb.
= a c c e l e r a t i o n due to g r a v i t y , 3 2 , 2 f t . / s e c .
The F r o u d e n u m b e r ,
Normally,
F,
2
1s defined by
the s q u a r e r o o t of the r a t i o of t h e i n e r t i a f o r c e s to the
g r a v i t y f o r c e s i s used a s the F r o u d e n u m b e r , thu.s
T h e length,
L,
channel flow,
i n the F r o u d e n u m b e r m a y be any lengch, bub m open
L i s u s u a l l y t a k e n t o be the depth of flow.
depth i s t a k e n a s the cross-sectional a r e a , A ,
the s u r f a c e width,
T h e hydrautac
of the w a t e r divided by
T ( A / T ) (Chow, 1959). F o r r e c t a n g u l a r c h a n n e l s
A / T would b e the depth of f l o w , but f o r t r a p e z o i d a l channels would b e
s o m e constant t i m e s the depth.
Hence, the F r o u d e n u m b e r s t o be
evaluated i n this study will u s e the hydraulrc depth @ I T ) f o r t h e iengch
measurement.
The F r o u d e n u m b e r will v a r y a t each c r o s s s e c t i o n i n
t h e f l u m e b e c a u s e the flow depth and m e a n velocity change f r o m c r o s s
s e c t i o n to c r o s s section.
Model m e a s u r e m e n t s a r e converted to corresponding prototype
m e a s u r e m e n t s by the l a w s of similitude.
u s e d to denote prototype p r o p e r t i e s ,
and " r "
The s u b s c r i p t "p"
will b e
"m" to denote m o d e l p r o p e r t i e s ,
to denote the r a t i o of t h e prototype p r o p e r t i e s t o t h e m o d e l
properties.
The fundamental r e q u i r e m e n t f o r the design of a F r o u d e model
i s t h a t the F r o u d e n u m b e r b e t h e s a m e in the m o d e l and in the prototype
(Murphy, 1950), thus F
model Froude number,
r
( r a t i o of prototype F r o u d e n u m b e r ,
F ) i s equal to one.
rn
A s mentioned b e f o r e , the length r a t i o i s equal to s e v e n .
F
P'
to
aq L e n
s e uaTJ:xmax
Z
'JJ
' M O ~ J JO
eaxe IeuoqJas- s s o x ~ =
v
=
0
'sas/
E
'75 %$ex
MOTJ
YJlYM
U!.
AV = 0
i(q;nu;$uo3 jo uo;qenba ayL
.
.
.
.
.
.
.
s 9 . z =LJ
'1 =
x
n
A59'Z=
=
B pue 1 =
I
d
A
.
I
3=
A
.I
J
asuIs 'pue
In the prototype s t r u c t u r e , pipes will b e extended into the flow,
both u p s t r e a m and d o w n s t r e a m f r o m the flume, to m e a s u r e the depths
of flow a t t h e s e l o c a t i o n s .
T h e s e pipes will l e a d to stilling wells p l a c e d
along s i d e the f l u m e , in which f l o a t s and a r e c o r d b r will b e located.
To duplicate t h i s condition in t h e m o d e l f l u m e , tubing w a s used to
m e a s u r e the u p s t r e a m and d o w n s t r e a m flow depth.
The u p s t r e a m a n d
d o w n s t r e a m depth m e a s u r e m e n t s , a s r e a d in the m o d e l stilling w e l l s ,
w e r e c o r r e l a t e d with the d i s c h a r g e r a t e s through the f l u m e to yleld the
n e c e s s a r y calibration.
The prototype f l u m e will b e c o n s t r u c t e d of c o n c r e t e . The a v e r a g e
r o u g h n e s s height f o r a c o n c r e t e s u r f a c e v a r i e s f r o m 0. 001 to 0. 01 feet
(King, 1954).
To obtain a n equivalent r o u g h n e s s in the model it would
b e n e c e s s a r y t o have a roughness height s o m e w h e r e between 0.0001 to
0. 001 f e e t .
T o obtain this roughness height, t h e m o d e l flume w a s
c o n s t r u c t e d of plywood with a sanded-painted s u r f a c e .
The e s t i m a t e d
r o u g h n e s s height w a s about 0 , 0 0 1 f e e t .
The initial d e s i g n used f o r the m o d e l t r a p e z o i d a l m e a s u r i n g
f l u m e i s shown i n F i g u r e 3 .
E X P E R I M E N T A L FACILITIES
A f t e r t h e c o n s t r u c t i o n of t h e m o d e l t r a p e z o i d a l m e a s u r i n g f l u m e
was completed
( F i g u r e 4 ),
i t w a s p r o p e r l y placed in t h e f i v e - f o o t
by f i v e - f o o t f l u m e ( F i g u r e 5 ), l o c a t e d i n t h e f l u i d m e c h a n i c s
laboratory.
T w o p u m p s w e r e u s e d , o p e r a t i n g t o g e t h e r and c a p a b l e of
delivering over 3 cfs.
T h e flow r a t e w a s r e g u l a t e d by m e a n s of a
v a l v e l o c a t e d on t h e l i n e a s i t e n t e r s the l a b o r a t o r y .
When s m a l l e r
f l o w s w e r e d e s i r e d i t w a s n e c e s s a r y t o u s e only one p u m p .
The water
w a s p u m p e d t h r o u g h a 1 2 - i n c h - d i a m e t e r pipeline which f e e d s into t h e
f i v e - f o o t b y five-foot f l u m e .
A t t h e beginning of t h i s f l u m e is a s c r e e n
w h i c h p r o v i d e s a n e v e n d i s t r i b u t i o n of the flow.
the f l u m e and discharged into weighing tanks.
T h e flow p a s s e d t h r o u g h
T h e flow r a t e w a s
c a l c u l a t e d f r o m t h e weight of w a t e r c a u g h t i n the t a n k s d u r i n g a
particular time period.
T h e w a t e r w a s w a s t e d f r o m the weighing t a n k s
into the s u m p , w h e r e i t w a s r e c i r c u l a t e d ( F i g u f e 7 ) .
When the flow w a s p a s s i n g t h r o u g h t h e t r a p e z o i d a l f l u m e , m e a s u r e m e n t s w e r e m a d e of ( 1 ) u p s t r e a m d e p t h , ( 2 ) m i n i m u m d e p t h in t h e t h r o a t
a n d i t s l o c a t i o n , ( 3 ) c r i t i c a l - d e p t h a n d i t s l o c a t i o n , and ( 4 ) d o w n s t r e a m
depth.
A l l d e p t h m e a s u r e m e n t s w e r e m a d e b y the u s e of a point g a g e , and
r e a d i n g s w e r e m a d e t o t h e n e a r e s t 0 . 0 0 1 foot.
C o p p e r tubing ( F i g u r e s
8 a n d 9 ) r u n n i n g f r o m u p s t r e a m and d o w n s t r e a m ends of the t r a p e z o i d a l fluEe
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~ S 30
M a y u r e a q s d n .6 a x n % ? j
into stilling w e l l s , located n e a r the m i d d l e of the flume ( F i g u r e 6),
provided f o r m e a s u r e m e n t of the u p s t r e a m and d o w n s t r e a m depths.
A
c r o s s b a r a c r o s s the flume w a s u s e d to s u p p o r t a point gate to m e a s u r e
the m i n i m u m and critical-depths o c c u r r i n g in t h e t h r o a t ( F i g u r e s
8 and 9 ) .
A t a i l g a t e w a s placed d o w n s t r e a m f r o m the dlume exit in o r d e r
to r e g u l a t e the tailwater depth c o r r e s p o n d i n g to that to be encountered
ln t h e field ( F i g u r e 1 0 ) .
DESIGN CI-IANGES RESULTING PROM MODEL STUDY
The m o s t s e v e r e conditon which the trapezoidal m e a s u r i n g f l u m e
m u s t undergo f r o m t h e standpoint of d o w n s t r e a m e r o s i o n and total
e n e r g y l o s s i s the d e s i g n flow of 300 c f s , whlch c o r r e s p o n d s to 2 . 3 1
c i s in the model.
Consequently, t h e effect of design c h a n g e s was
evaluated i n t h e m o d e l a t 2 31 c f s .
A f t e r design flow w a s e s t a b l i s h e d i n the model, the t a i l w a t e r w a s
a d j u s t e d t o the height which gave the existing t a i l w a t e r d e p t h t o
be encountered in the f i e l d , which f o r the model a t d e s i g n flow w a s
0. 57 feet.
With the w a t e r flowing a t d e s i g n conditions, o r a t any o t h e r flow,
s e p a r a t i o n of flow o c c u r r e d a t the end of the throat.
The s k i n f r i c t i o n
of the f l u m e w a s n ' t g r e a t enough to c a u s e the flow to d e c e l e r a t e rapidly
enough to exit uniformly, b e c a u s e the divergence of the exit s e c t i o n w a s
too g r e a t .
This s e p a r a t i o n c a u s e d t h e flow to a d h e r e to one s i d e o r the
o t h e r of the diverging wall. (See F i g u r e s 11 and 12.3 A r e c i r c u l a t i o n
o c c u r r e d d o w n s t r e a m allowing p a r t of the flow to c o m e back t o the exit
on the opposite s i d e of the f l u m e f r o m which it was leaving.
The m a i n
r e a s o n f o r the s e p a r a t i o n i s t h e inability of the flow to d i v e r g e a t t h e
a n g l e t h a t the f l u m e w a s c o n s t r u c t e d , which w a s
included angle of this d i v e r g e n c e i s 3 6 . 9
0
.
t h r e e t o one.
The
F r o m Chqw (1959, p. 3141,
the length of a t r a n s i t i o n should b e d e t e r m i n e d s o that
"a s t r a i g h t line
joining the flow 1lne a t the two ends of the t r a n s i t ~ o nwill m a k e a n angle
of about 1 2 . 5' with the a x i s of t h e s t r u c t u r e . "
Similar c r i t e r i a a r e
given by Hinds (1928) c o n c e r n i n g the m a x i m u m d i v e r g e n c e angle.
T h i s condition would r e q u i r e a divergence of about 9 : 1 i n s t e a d of 3 : 1 .
To e l i m i n a t e s e p a r a t i o n of flow and to b r e a k up the jet leaving
the t h r o a t , s e v e r a l m e t h o d s w e r e explored.
Vanes c o n s t r u c t e d of s t e e l
w e r e placed on the t r a p e z o i d a l f l u m e bottom in the exit s e c t i o n s i n
an a t t e m p t t o d i s t r i b u t e the flow evenly.
T h e s e v a n e s w e r e placed in
v a r i o u s p a t t e r n s , with s o m e t r i a l s extending v a n e s up into the f l u m e
throat.
The v a n e s w e r e m a d e of varying heights and s i z e s .
T h e r e was
n o noticeable effect of the v a n e s upon the m a i n flow conditions.
Critical-
depth always o c c u r r e d in t h e t h r o a t and a t t h e s a m e l o c a t i o n f o r the.
d e s i g n flow,
The head l o s s a l s o r e m a i n e d about the s a m e ,
p r o b l e m of s e p a r a t i o n w a s s t i l l p r e s e n t .
The
(See F i g u r e s 1 3 and 14. )
Next, wooden blocks w e r e placed in the Plume bottom.
Various
p a t t e r n s and a r r a n g e m e n t s w e r e used.lSee F i g u r e 15, 16, 1 7 , and 18,)
T h e s e blocks w e r e always placed with the idea i n mind t h a t t h e y w o u l d
not t r a p p a s s i n g s e d i m e n t but allow it to p a s s through the blocks.
Double r o w s of blocks w e r e t r i e d
.
( F i g u r e s 1 7 and 18)
with the
second row s t a g g e r e d behind the f i r s t a s well a s the second row being
placed a t v a r y i n g d i s t a n c e s behind t h e f i r s t .
Using the b l o c k s , the flow
of w a t e r w a s d i f f e r e n t depending on the p a t t e r n u s e d .
The c l o s e r t o the
t h r o a t the blocks w e r e placed, the g r e a t e r t h e d e g r e e of s u b m e r g e n c e
that o c c u r r e d i n the t h r o a t .
Although the blocks b r o k e up the jet and p a r t i a l l y prevented
s e p a r a t i o n , and although s o m e a r r a n g e m e n t s ( t h o s e placed f u r t h e r
f r o m the t h r o a t ) evenly distributed the flow a c r o s s the exit, the m a i n
objection t o t h e i r u s e w a s t h e c r e a t i o n of a h y d r a u l i c jump ( F i g u r e
16) d i r e c t l y behind the b l o c k s , yielding a w o r s e condition f o r e r o s i o n
a t the end of t h e f l u m e .
Another f e a t u r e which d i s c o u r a g e d the u s e of
the blocks w a s t h e s u b m e r g e d flow t h e r e b y c r e a t e d i n the t h r o a t , which
prevented o p e r a t i o n a s a c r i t i c a l - d e p t h f l u m e .
A f t e r m a n y t r i a l s , it w a s concluded that t h e exiting section would
have to b e c o n s t r u c t e d with a divergence m u c h l e s s than the one being
used.
F r o m o t h e r s t u d i e s , t h e i d e a l d i v e r g e n c e should b e 9 ; 1, bnt
in view of economic f a c t o r s i t w a s decided t o d i v e r g e a t 6 : 1.
B e f o r e a new exit section (Section 6 ) w a s c o n s t r u c t e d , the f i r s t
converging s e c t i o n (Section 1) was r e m o v e d , a s it s e e m e d f r o m o h s e r vation of the e n t e r i n g flow that t h i s s e c t i o n m i g h t possibly be eliminated.
With Section 1 r e m o v e d , t h e f l u m e w a s t e s t e d u n d e r design conditions
':eo.zya aya 30 pua ae
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and v e r y l i t t l e d i f f e r e n c e w a s observed in t h e e n t e r i n g flow conditions
o r in the m e a s u r e m e n t s .
in the t h r o a t .
C r i t i c a l - d e p t h o c c u r r e d a t the s a m e location
The flow e n t e r e d the flume smoothly ( F i g u r e 19) and
with a slight c o n t r a c t i o n a t the two s i d e s .
The amount of h e a d l o s s
through t h e f l u m e , with Section 1 r e m o v e d , w a s r e d u c e d by 0 . 0 0 5 f e e t .
Economically then, i t w a s f e a s i b l e to r e m o v e the f i r s t section s i n c e
d e s i r a b l e e n t r a n c e conditions could b e maintained.
The 6 : 1 d i v e r g e n c e section w a s c o n s t r u c t e d inside of the
d i s c a r d e d 3 : 1 exit
( F i g u r e s 2 0 and 21).
The construction and
flnish w a s done in the s a m e m a n n e r a s the 3 : 1 e x i t s e c t i o n had b e e n .
With the w a t e r flowing a t design conditions, s e p a r a t i o n of flow
still o c c u r r e d .
The p r o b l e m w a s again approached u s i n g v a n e s
(Flgure 22) and b l o c k s , and with the s a m e r e s u l t s a s previously
attained.
A d e s i g n w a s then t r i e d using wooden blocks ( e s s e n t i a l l y c o l u m n s )
high enough to p r o t r u d e above the w a t e r s u r f a c e a t t h e m a x i m u m flow.
In the beginning t h e s e columns w e r e used in conjunction with the s t e e l
vanes
( F i g u r e s 2 3 and 2 4 ) .
s e e n in t h e s e f i g u r e s ,
Some idea of the p a t t e r n s t r i e d can be
T h e r e w a s s o m e l i m i t e d s u c c e s s in the u s e of
two columns ( F i g u r e 24) m distributing t h e flow a l i t t l e m o r e
evenly.
The next s t e p , then, w a s to t r y v a r i o u s p a t t e r n s of columns.
The columns w e r e u s e d in one and two rows and with v a r i e d spacing
b e f o r e the b e s t p o s s i b l e combination was s e l e c t e d .
F i g u r e s 2 5 and 2 6
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MOYS
ANALYSIS O F DATA FROM FINAL DESIGN
Super-critical flow a n a l y s i s
The m o s t d e s i r a b l e condition f o r predicting the d i s c h a r g e i n a
f l u m e i s t o have t h e t h r o a t sufficiently c o n s t r i c t e d t o p r o d u c e c r i t i c a l depth,
When c r i t i c a l - d e p t h o c c u r s in the t h r o a t , the only m e a s u r e m e n t
r e q u i r e d t o d e t e r m i n e t h e d i s c h a r g e , through t h e f l u m e , i s the u p s t r e a m
depth.
T h e proposed prototype d e s i g n w a s c a l i b r a t e d a s a c r i t i c a l - d e p t h
f l u m e by m e a n s of a m o d e l ,
previously s t a t e d .
The flow and depth w e r e m e a s u r e d a s
A s long a s the flow p a s s e d through c r i t i c a l - d e p t h
in the t h r o a t , the flow w a s independent of the d o w n s t r e a m depth even
though the u p s t r e a m depth v a r i e d with d i s c h a r g e .
A f t e r the d a t a were r e c o r d e d , the flow w a s plotted a s t h e o r d i n a t e
and the depth a s the a b s c i s s a .
A conversion was made t o corresponding
prototype m e a s u r e m e n t s and the data w e r e plotted on both c a r t e s i a n and
log-log p a p e r .
On log-log p a p e r , the points plotted a s a s t r a i g h t l i n e
( F i g u r e 33 ) which shows tha$ a n exponential r e l a t i o n s h i p e x i s t s
between flow and depth.
The equation of a s t r a i g h t l i n e on l o g - l o g p a p e r
c a n be w r i t t e n a s
l o g y = s (log x)
+ log C
o r y = Cx
S
in which
y
=
the function plotted a s the ordinate
x
=
the function plotted on the a b s c i s s a
s
=
the slope of the line
C
=
constant equal to the value of y when x = 1. 0
thus
log Q = s l o g (h 3
P
1P
+ log C
or
From F i g u r e 33
s
=
1,78
C
=
18.0
therefore
The corresponding equation relating depth and flow in the model,
which w a s obtained i n the s a m e way, i s
Equation 16 allows the calculation of any flow passing through the
prototype flume provided the u p s t r e a m depth i s m e a s u r e d and c r i t i c a l depth h a s o c c u r r e d in the t h r o a t .
S u b m e r g e d flow a n a l y s i s
If a n y flow p a s s i n g t h r o u g h t h e f l u m e f a i l s t o p a s s t h r o u g h c r i t i c a l d e p t h , the flow is s a i d t o b e s u b m e r g e d .
S u b m e r g e n c e , a s defined b y
Wells a n d G o t a a s ( 1 9 4 8 ) , i s ' I t he p e r c e n t r a t i o of t a i l w a t e r d e p t h t o t h e
u p s t r e a m d e p t h of flow w h e r e the t a i l w a t e r d e p t h i s r e f e r r e d t o t h e
channel i n v e r t a t t h e point of u p s t r e a m m e a s u r e m e n t .
"
Many a s p e c t s of s u b m e r g e n c e r e s e a r c h h a v e not yet b e e n s t u d i e d .
One a p p r o a c h t o t h e s i t u a t i o n i s t h a t m a d e b y R o b i n s o n (1964) w h e r e
( h / h ) is p l o t t e d a g a i n s t Q / Q , i n which
0
4 1
Q
Q
0
=
observed discharge
=
t h e o r e t i c a l d i s c h a r g e obtained f r o m E q u a t i o n 7
Robinson (1964) found t h a t the flow did not r e q u i r e c o r r e c t i o n
until the s u b m e r g e n c e r e a c h e d a p p r o x i m a t e l y 80 p e r c e n t .
Then, f o r
s u b m e r g e n c e g r e a t e r t h a n 80 p e r c e n t , the plot of s u b m e r g e n c e v e r s u s
Q/Q
0
can b e utilized to obtain Q / Q
0
.
The theoretical discharge,
c a n b e obtained f r o m t h e u p s t r e a m d e p t h ,
Q,
hl,
Qo,
and thus the discharge
can b e computed.
T h e d a t a f r o m t h i s r e s e a r c h effort h a v e b e e n p l o t t e d i n , t h e s a m e
m a n n e r a s u s e d b y R o b i n s o n ( F i g u r e 35).
This r e s e a r c h effort also
found t h a t n o c o r r e c t i o n f o r flow w a s n e e d e d u n t i l a s u b m e r g e n c e of
a p p r o x i m a t e l y 80 p e r c e n t w a s r e a c h e d a n d the plot of ( Q / Q ) v e r s u s
0
jh4/hl) w a s identical in shape to those produced b y Rdbinson (1964).
F r o m F i g u r e 35, i t c a n b e s e e n that c o n s i d e r a b l e s c a t t e r e x i s t s
in the d a t a .
The u s e of F i g u r e 35 f o r s u b m e r g e d flow conditions would
r e s u l t in c o n s i d e r a b l e e r r o r in d e t e r m i n i n g the d i s c h a r g e ,
Consequently,
it w a s f e l t that a n i m p r o v e d method of analyzing s u b m e r g e d flow in a
trapezoidal flume was n e c e s s a r y .
C o n s i d e r a b l e thought w a s given to a n a p p r o a c h f o r analyzing
s u b m e r g e d flow conditions.
The submergence,
to be an a p p r o p r i a t e p a r a m e t e r .
h 4 / h l , was considered
The proper c r i t e r i a for super-critical
o r s u b c r i t i c a l flow in t h e t h r o a t i s the F r o u d e n u m b e r .
Consequently,
the F r o u d e n u m b e r w a s evaluated a t the c r o s s s e c t i o n of the t h r o a t
w h e r e m i n i m u m depth o c c u r r e d .
This F r o u d e number,
Fmax,
i s actually t h e m a x i m u m F r o u d e n u m b e r o c c u r r i n g in the f l u m e .
The
o t h e r p a r a m e t e r , which will b e r e f e r r e d to a s t h e e n e r g y . l o s s
p a r a m e t e r , i s defined a s (h
1
- h ) / h . The energy l o s s parameter
4
m
-
w a s s e l e c t e d a s a m e a n s of using the energy . l o s s ,
h
1
- h
4'
as a
significant p a r a m e t e r m a d e d i m e n s i o n l e s s by division of the minimum
depth of flow in t h e t h r o a t , h
m
-
, also
m
-
The u s e of m i n i m u m depth, h
had the advantage of utilizing the conditions a t t h e t h r e e i m p o r t a n t
c r o s s sections.
The p a r a m e t e r s involved in s u b m e r g e d flow i n t r a p e z o i d a l
m e a s u r i n g f l u m e s c a n b e obtained f r o m d i m e n s i o n a l a n a l y s i s , a s
follows :
With five independent quantities and two d i m e n s i o n s , t h r e e p i - t e r m s
a r e derived,
Equatxon 19 c a n be modified by replacing V with Q / A
m
-
/T .
m
m
-
and h
m
-
A
TI
1
Q
=
-
-
Am ' g A m /
= F
T
max
m
-
in which
Q
= flow r a t e , cfs
A m = m i n i m u m a r e a , equals ( b t hm ) b, f t .
2
b
= f l u m e t h r o a t bottom width, f t
h
= m i n i m u m depth of flow in the t h r o a t , f t .
m
-
T
g
m
= m i n i m u m w a t e r s u r f a c e width of flow in the t h r o a t , i t
= a c c e l e r a t i o n due to g r a v i t y , 32. 2 f t . / s e c .
2
with
T h e relationship between s u b m e r g e n c e and the e n e r g y l o s s
p a r a m e t e r w a s developed by plotting the log of s u b m e r g e n c e a s the
o r d i n a t e and the energy l o s s p a r a m e t e r a s the a b s c i s s a .
The r e l a t i o n -
s h i p w a s e s s e n t i a l l y a s t r a i g h t l i n e ( F i g u r e 36) which can b e w r i t t e n a s
a n equation
o r simplifying
0.99
=
Submergence = h / h
4 1
.
.
.
23
0.34
10
A log-log plot w a s p r e p a r e d between the energy l o s s p a r a m e t e r
and t h e m a x i m u m F r o u d e n u m b e r ,
F
max'
( F i g u r e 37).
The energy
l o s s p a r a m e t e r w a s plotted a s the ordinate and F
w a s plotted a s
max
the a b s c i s s a .
The relationship w a s e s s e n t i a l l y a s t r a i g h t l i n e and
r e s u l t e d in the equation
To show the relationship between the t h r e e p i - t e r m s
h 4 / h l , and (h
1
-
F
max'
h 4 ) / h m a n additional plot w a s m a d e between s u b -
-
m e r g e n c e and the e n e r g y l o s s p a r a m e t e r .
The e n e r g y l o s s p a r a m e t e r
w a s plotted on the log s c a l e a s t h e ordinate and to the s a m e s c a l e a s in
F i g u r e 37.
scale.
Submergence w a s plotted a s the a b s c i s s a on a c a r t e s i a n
T h i s plot ( F i g u r e 37) yields a p r a c t i c a l graphical solution of
-mu
30 801 pue ;raqaurexed s s o i(8;raua
~
uaamqaq drysuoqe1a\6
.9g a ; r n 8 1 ~
F
max
when the s u b m e r g e n c e i s known a s well a s showing the r e l a t i o n -
ship between the t h r e e p a r a m e t e r s o r p i - t e r m s ,
With the r e l a t i o n s h i p between s u b m e r g e n c e and F r o u d e n u m b e r
known, i t w a s d e s i r e d to r e l a t e t h e s e two p a r a m e t e r s to d i s c h a r g e .
F i r s t , a log-log t h r e e - d i m e n s i o n a l plot of h
(hl
m'
-
- h4)/hm
-
was prepared.
a s the o r d i n a t e , h
m
-
(hl
- h4),
and
The energy l o s s p a r a m e t e r w a s plotted
a s t h e a b s c i s s a , and h
For the f a m i l y of c u r v e s s e e F i g u r e 38.
a s t h e variable,
1 - h4
Since h
and h 4 a r e
1
m e a s u r e d , the value of the e n e r g y l o s s p a r a m e t e r c a n b e obtained f r o m
F i g u r e 37, t h e r e b y allowing t h e determination of h
m
-
f r o m F i g u r e 38.
, F
Next, a t h r e e - d i m e n s i o n a l log-log plot w a s m a d e of h
Q.
and d i s c h a r g e ,
Here h
m
-
w a s plotted a s the o r d i n a t e ,
m
-
max'
F
as
max
the a b s c i s s a , and d i s c h a r g e a s the p a r a m e t e r f o r t h e f a m i l y of c u r v e s
( F i g u s e 39. )
The solution f o r any d i s c h a r g e , given the u p s t r e a m
and d o w n s t r e a m d e p t h s , would entail the u s e of F i g u r e 38 to obtain a
value of h
; the u s e of F i g u r e 37 to obtain the F r o u d e n u m b e r ; and
m
-
then f r o m F i g u r e 39 a value of d i s c h a r g e could b e i n t e r p o l a t e d .
However, a n equation f o r evaluating the d i s c h a r g e f r o m F i g u r e
39 can b e obtained b y w r i t i n g the equation of e a c h of t h e l i n e s ,
h
m
-
= C F
5 max
The coefficient,
a value of C
5
C5'
S
i s the value of h
f o r F = 1. 0.
m
-
Consequently,
i s obtained f o r each line of constant d i s c h a r g e ( s e e the
t a b l e included in F i g u r e 39). A log-log plot was t h e n p r e p a r e d between
.a8sey3srp pue ' j e o x y ~u!
yJdap urnru!urur 'xaqrunu apnoL3 uaam3aq d : y s u o ~ ~ e ~' a6 ~a
~x n 8 l s
the p a r a m e t e r C
and d i s c h a r g e
5
r e l a t i o n s h i p between d i s c h a r g e ,
Q = 34.7 C
1.74
h
m
-
C
5
h
and F
max
m
-
= C F
5 max
and C
5
straight-line
can be expressed b y
a r e related to C
5
25
as
-0.57
0.57
=h F
Q,
The
. . . , . . . . . . . .
5
F r o m F i g u r e 39,
( F i g u r e 40).
. . . . . . . . . .
m max
-
Combining Equations 25 and 26
Q = 34.7 F
h
1.74
-
max m
. . . . . . . . . .
To obtain the r e l a t i o n s h i p between Q and h / h
4 1'
27
Equations 27
and 24 a r e combined t o yield
which, when combined with Equation 23 and s i m p l i f i e d , yields
-13.83 ( h
Q=
1
- h
)
1.74
4
. . . . ? . . .
-h4
1.32
(log- t 0. 0044)
1
Although Equation 28 i s only valid f o r the t r a p e z o i d a l m e a s u r i n g
f l u m e s t u d i e d , i t d o e s show t h a t only the u p s t r e a m and d o w n s t r e a m
depths need t o b e m e a s u r e d t o d e t e r m i n e t h e d i s c h a r g e under subm e r g e d flow conditions in any t r a p e z o ~ d a fl l u m e .
'Jeo.zy7 ur y ~ d a purnurIuIur pue 'xaqurnu apno.zA
'a2.z.ey3sjp uaamJaq dyysrtoq~1a.x30 p a u t d o ~ a ~ a a
.op a.zn8rj
In o r d e r to p e r f o r m t h e a r i t h m e t i c operations of the equations f o r
the submerged trapezoidal flume, a computer program was written.
T h i s p r o g r a m , including input and output d a t a , i s l i s t e d in Appendix B.
The computations show t h e a c c u r a c y of Equations 23, 24, and 28.
The a c c u r a c y obtained i n calculating d i s c h a r g e a s a function of s u b m e r g e n c e only i s of s p e c i a l i n t e r e s t .
The computer p r o g r a m output
showed that the d i s c h a r g e a s computed f r o m Equation 28 h a s a p e r c e n t a g e of e r r o r ( b a s e d on m e a s u r e d v a l u e s ) of 0 . 5 0 p e r c e n t f o r s u b m e r g e n c e values g r e a t e r than 90 p e r c e n t ; 1,78 p e r c e n t f o r s u b m e r g e n c e
v a l u e s between 85 to 9 0 p e r c e n t ; and 3 . 8 2 p e r c e n t f o r s u b m e r g e n c e
v a l u e s f r o m 85 p e r c e n t to c r i t i c a l flow.
The a v e r a g e e r r o r f o r all
m e a s u r e d values of s u b m e r g e n c e w a s 1 . 4 1 p e r c e n t .
Hence, the
r e l a t i o n s h i p obtained between d i s c h a r g e and s u b m e r g e n c e i s m o r e
a c c u r a t e f o r the higher s u b m e r g e n c e v a l u e s .
T h e relationships thus a r r i v e d a t in t h e preceding equations a r e
valid and the d e g r e e of i n a c c u r a c y i s p r i m a r i l y due to e x p e r i m e n t a l
procedures.
The amount of e r r o r i s s a t i s f a c t o r y f o r m o s t f i e l d flow
m e a s u r e m e n t stations.
One of the p r i m a r y p u r p o s e s of t h i s investigation h a s b e e n t h e
d e s i g n and c a l i b r a t i o n of a prototype t r a p e z o i d a l m e a s u r i n g f l u m e which
could b e c o n s t r u c t e d in Canal
33,M . A . D .
Company.
"B" of
the distribution s y s t e m of t h e
P r i m a r i l y , t h e f l u m e will o p e r a t e with c r i t i c a l -
d e p t h o c c u r r i n g in the t h r o a t and consequently, only the u p s t r e a m depth,
.asuaE!;cartrqhs pu-e ' s s o ~AE!.zaua
'a8;c-eyssrp uaamJaq d?ysuor~eljax30 $ u a u r d o ~ a n a a . I * a z n 8 1 ~
LEI
hl,
If the t a i l -
will have to b e m e a s u r e d t o d e t e r m i n e the d i s c h a r g e .
w a t e r depths in Canal "B" should i n c r e a s e in the f u t u r e , t h e f l u m e m a y
become submerged.
T h e r e f o r e , c a l i b r a t i o n c u r v e s f o r s u b m e r g e d flow
have been p r e p a r e d .
In o r d e r to p r e p a r e c a l i b r a t i o n c u r v e s f o r s u b m e r g e d flow, a
t h r e e - d i m e n s i o n a l log-log plot w a s p r e p a r e d of Q,
h4/hl.
h
1
- h
The d i s c h a r g e ,
4'
Q,
h
1
- h
4'
and
w a s plotted a s the ordinate, e n e r g y l o s s ,
a s the a b s c i s s a , and s u b m e r g e n c e ,
For the f a m i l y of c u r v e s s e e F i g u r e 41.
h4/hl,
a s the variable.
A s e r i e s of p a r a l l e l l i n e s of
v a r y i n g s u b m e r g e n c e w e r e then d r a w n f o r s u b m e r g e n c e s b e t w e e n 80
p e r c e n t and 96 percent.(See F i g u r e 41.3 In t h e field then, f o r a m e a s u r e d u p s t r e a m and d o w n s t r e a m depth, the energy l o s s ,
the submergence, h 4 / h l ,
h
1
- h4 '
and
c a n b e computed, thus allowing a d e t e r -
m i n a t i o n of the d i s c h a r g e f r o m F i g u r e 42 f o r the prototype t r a p e z o i d a l
measuring flume.
E N E R G Y LOSSES I N MEASURING F L U M E S
A c o m p a r i s o n of the e n e r g y l o s s e s in t h r e e t y p e s of v e n t u r i
f l u m e s ( P a r s h a l l , r e c t a n g u l a r , a n d t r a p e z o i d a l ) is t o b e m a d e .
These
flumes a r e all v e n t u r i f l u m e s w h i c h c o n s i s t of a g r a d u a l l y c o n v e r g i n g
portion (the e n t r a n c e ) , a constricted portion (the t h r o a t ) , and a gradually
diverging portion (the exit).
T h e c o m p a r i s o n of t h e f l u m e s w i l l be b a s e d
upon d e s i g n c o n d i t i o n s f o r C a n a l
"B"w h e r e the h e a d l o s s t h r o u g h e a c h
t y p e f l u m e m u s t b e l e s s t h a n one foot and the m a x i m u m d i s c h a r g e i s
300 c f s .
Under d e s i g n c o n d i t i o n s , m o d e l m e a s u r e m e n t s c o n v e r t e d t o
p r o t o t y p e m e a s u r e m e n t s f o r t h e t r a p e z o i d a l f l u m e gave a n u p s t r e a m
d e p t h e q u a l t o 4. 865 f e e t w i t h a d o w n s t r e a m d e p t h of 3 , 9 5 5 f e e t .
T h e r e f o r e , t h e h e a d l o s s t h r o u g h the f l u m e i s 0 . 9 1 f e e t .
A prototype r e c t a n g u l a r m e a s u r i n g flume w a s designed according
t o t h e c r i t e r i a of Kinfg (1954),
Q
=
c M a 2 4h
. . .
.
.
.
.
.
.
.
.
.
.
29
i n which
=
c
constant equal to 0.95
-
1. 0 and t a k e n , c o n s e r v a t i v e l y , a s
0.95.
a
h
2
2
=
a r e a a t throat, ft.
=
d i f f e r e n c e i n w a t e r l e v e l s a t the e n t r a n c e a n d t h r o a t , f t .
Q
= f l o w r a t e , cfs.
a
=
1
a r e a at entrance, ft.
2
F o r a flow r a t e of 300 c f s , and using c = 0 . 9 5 ,
the t h r o a t width of
10 f e e t gives the s a m e total e n e r g y (4. 75) a s computed f o r the t r a p e zoidal f l u m e .
F o r the prototype r e c t a n g u l a r f l u m e design s e e F i g u r e
43.
A prototype P a r s h a l l m e a s u r i n g f l u m e w a s designed a c c o r d i n g to
the c r i t e r i a of the U . S. B u r e a u of R e c l a m a t i o n (1953) in which
in which
Q
=
discharge, cis.
W
=
width of t h r o a t , f t .
H
=
upper h e a d , f t .
a
The value of H
a
w a s obtained f r o m I s r a e l s e n (1953) in which the
m i n i m u m head l o s s f o r f r e e flow equals 0.4 x H , and the m i n i m u m
a
head loss w a s t a k e n a s 1 . 0 f e e t f o r a c o m p a r i s o n with the t r a p e z o i d a l
flume.
The d e s i g n w a s then completed a c c o r d i n g t o t h e U. S . B u r e a u
of Reclamation (1953) and F i g u r e 44 shows t h e d e t a i l s of the design.
The c o m p a r i s o n of the s t r u c t u r e s , a l l of which w i l l a c c o m p l i s h the
s a m e o p e r a t i o n with the s a m e head l o s s , shows that 46. 5 cubic y a r d s
of c o n c r e t e a r e n e c e s s a r y f o r construction of t h e t r a p e z o i d a l f l u m e ,
45 cubic y a r d s f o r the r e c t a n g u l a r , and 3 3 . 5 cubic y a r d s of c o n c r e t e
a r e needed to e r e c t t h e P a r s h a l l f l u m e .
F i g u r e 43.
END
Design of r e c t a n g u l a r m e a s u r i n g flu*----------------
VIEW
F i g u r e 44.
Design of Parshall m e a s u r i n g f l u m e .
E v e n though the P a r s h a l l flume r e q u i r e s a s m a l l e r volume of
c o n c r e t e it h a s two m a i n disadvantages b e c a u s e of t h e n e c e s s i t y to s e t
the f l o o r of the s t r u c t u r e a t a n elevation which wlll sirtisfy the n o r m a l
headwater conditions.
T h e s e disadvantages are: ( 1) s i l t depositing
u p s t r e a m , and ( 2 ) i n c r e a s e d depth u p s t r e a m , c a u s i n g additional s e e p age l o s s e s .
Canal
"B" i s v e r y s i m i l a r to C a n a l "A".
In Canal "A",
m e a s u r e m e n t s showed t h a t a n i n c r e a s e i n flow depth of 2 . 3 f e e t caused
an additional s e e p a g e l o s s of m o r e than 20 c f s .
At 20 c f s , the prototype
t r a p e z o i d a l m e a s u r i n g f l u m e h a s an u p s t r e a m d e p t h of 1 . 2 7 feet.
The
P a r s h a l l f l u m e with t h e s a m e flow r a t e h a s a n u p s t r e a m depth of 3. 02
feet.
Thus, t h e s e e p a g e l o s s e s would definitely i n c r e a s e with u s e of
the P a r s h a l l flume.
Under s u p e r - c r i t i c a l flow conditions, t h e t r a p e z o i d a l f l u m e and
the r e c t a n g u l a r f l u m e a r e v e r y c o m p a r a b l e with l i t t l e d i f f e r e n c e in the
operation of e i t h e r .
The m o d e l study of the t r a p e z o i d a l m e a s u r i n g f l u m e h a s shown
that such a f l u m e designed f o r submerged flow conditions c a n b e used
a s a measuring device.
A s u b m e r g e d flume h a s s e v e r a l advantages in
f a v o r of i t s u s e : (1) l e s s head l o s s through the s t r u c t u r e , (2) l e s s
e r o s i o n d o w n s t r e a m f r o m the s t r u c t u r e , ( 3 ) u s e of a s h o r t e r s t r u c t u r e
b e c a u s e of a f a s t e r exit d i v e r g e n c e .
The difficulty in c o m p a r i n g
different types of v e n t u r i f l u m e s under s u b m e r g e d flow conditions i s
the v e r y m e a g e r amount of i n f o r m a t i o n which e x i s t s .
Some of the
f o r m u l a s that have been cited a r e i n c o n s i d e r a b l e e r r o r and a r e e n t i r e l y
inadequate a s a b a s i s f o r designing a flow m e a s u r e m e n t s t r u c t u r e .
Both the r e c t a n g u l a r and t r a p e z o i d a l flat-bottomed flumes would b e
v e r y advantageous under s u b m e r g e d flow conditions.
It i s f e l t ,
h o w e v e r , that the t r a p e z o i d a l f l u m e , b e c a u s e of i t s g e o m e t r y , h a s a
slight advantage
over the r e c t a n g u l a r f l u m e .
SUMMARY AND CONCLUSIONS
In o r d e r to provide an e c o n o m i c a l flow m e a s u r i n g s t r u c t u r e f o r
t h e D. M. A. D. Company, and t o c o r r e l a t e this study with p r e v i o u s
r e s e a r c h , a model study w a s conducted of a trapezoidal m e a s u r i n g
flume.
T h e object of t h i s study w a s to provide the d e s i g n f o r a
prototype s t r u c t u r e c a p a b l e of c a r r y i n g 300 c f s and to c a l i b r a t e the
s t r u c t u r e f o r both s u p e r - c r i t i c a l and submerged flow conditions,
Utilizing r e s e a r c h e f f o r t s of o t h e r s , and the known field cond i t i o n s , a n initial d e s i g n f o r a prototype s t r u c t u r e w a s p r o p o s e d a n d
then a m o d e l study w a s m a d e of t h i s proposed design.
The r a t i o of
prototype to m o d e l w a s taken a s 7 : 1 in view of l a b o r a t o r y f a c i l i t i e s .
The initial design p r o p o s e d w a s not s u c c e s s f u l due to t h e
o c c u r r e n c e of s e p a r a t i o n , which w a s c a u s e d by too g r e a t a d i v e r g e n c e
i n t h e exit section,
T o r e c t i f y t h i s p r o b l e m , s e v e r a l p a t t e r n s of b l o c k s ,
s t e e l v a n e s , and columns w e r e u s e d to eliminate t h e s e p a r a t i o n and to
evenly d i s t r i b u t e the flow.
. T h i s proved i m p r a c t i c a l o n the i n i t i a l
3 : 1 diverging s e c t i o n , b u t on t h e next selection of a
6 : 1 diver-
g e n c e , a p a t t e r n of t h r e e c o l u m n s placed a t the beginning of the exit
and of height g r e a t e r than t h e m a x i m u m flow, fulfilled the r e q u i r e ments.
E x p e r i m e n t a t i o n a l s o showed that elimination of a p o r t i o n of
t h e e n t r a n c e section w a s justifiable.
The modified m o d e l w a s then c a l i b r a t e d and t h u s provided the
design of a prototype s t r u c t u r e f o r field u s e .
A r a t i n g c u r v e and
equation w e r e then developed f o r s u p e r - c r i t i c a l flow through the
prototype s t r u c t u r e .
By u s e of this c u r v e , the D . M , A , D. Company can
c o n s t r u c t and u s e t h i s t r a p e z o i d a l m e a s u r i n g f l u m e .
Should s u b m e r g e d flow develop i n the f i e l d , but p r i m a r i l y a s a
point of p r a c t i c a l i n t e r e s t , the prototype s t r u c t u r e w a s c a l i b r a t e d f o r
s u b m e r g e d flow by r e l a t i n g the m a x i m u m F r o u d e n u m b e r and subm e r g e n c e conditions to a new p a r a m e t e r , called t h e e n e r g y l o s s
parameter.
T h i s p a r a m e t e r i s actually the e n e r g y l o s s divided by the
m i n i m u m depth of flow in the t h r o a t .
T h i s r e l a t i o n s h i p yielded a r a t i n g
c u r v e and an equation f o r s u b m e r g e d flow conditions.
T o show the
a c c u r a c y of the r e l a t i o n s h i p a computer p r o g r a m w a s w r i t t e n .
The
computer p r o g r a m showed the a v e r a g e e r r o r f o r a l l m e a s u r e d values
of s u b m e r g e n c e to b e 1 . 4 4 p e r c e n t .
A c o m p a r i s o n w a s then m a d e of t h r e e v e n t u r i f l u m e s : (1)
trapezoidal, ( 2 ) r e c t a n g u l a r , and ( 3 ) P a r s h a l l .
These t h r e e flumes
w e r e s o designed that e a c h had one foot head l o s s through the s t r u c t u r e
f o r a d i s c h a r g e of 300 c f s .
The c o m p a r i s o n m a d e is the volume of
c o n c r e t e n e c e s s a r y to c o n s t r u c t each s t r u c t u r e .
The r e c t a n g u l a r and
t r a p e z o i d a l a r e e s s e n t i a l l y the s a m e but the P a r s h a l l r e q u i r e s l e s s
concrete.
However, it should be noted h e r e that s i l t i n g u p s t r e a m and
additional s e e p a g e l o s s e s d e c r e a s e d the d e s i r a b i l i t y of the P a r s h a l l
flume.
This investigation has provided a n adequate method f o r the
c a l i b r a t i o n of open channel s u b m e r g e d m e a s u r i n g f l u m e s .
The
p a r a m e t e r s n e c e s s a r y to c a l i b r a t e a t r a p e z o i d a l m e a s u r i n g f l u m e w e r e
developed.
Subsequent t e s t i n g by Skogerboe, W a l k e r , and Roblnson
(1965) h a s indicated that the s a m e p a r a m e t e r s a r e a l s o valid f o r
rectangular flumes.
The development of t h e s e p a r a m e t e r s will provide
the b a s i s f o r a m o r e w i d e s p r e a d u s e of s u b m e r g e d f l u m e s i n t h e f u t u r e .
S E L E C T E D BIBLIOGRAPHY
1.
A c k e r s , P . , and A. J. M . H a r r i s o n . 1963. C r i t i c a l - d e p t h
f l u m e s f o r flow m e a s u r e m e n t s in o p e n c h a n n e l s . H y d r a u l i c
R e s e a r c h P a p e r No. 5, H y d r a u l i c s R e s e a r c h S t a t i o n , D e p a r t m e n t
of S c i e n t i f i c and I n d i s t r i a l R e s e a r c h , Wallingford, B e r k s h i r e ,
England. April.
2.
Chow, V . T . 1959. O p e n c h a n n e l h y d r a u l i c s . M c G r a w - H i l l
B o o k C o m p a n y , I n c , , New Y o r k , New Y o r k , p. 63-85.
3.
H i n d s , J u l i a n . 1928. T h e h y d r a u l i c d e s i g n of f l u m e a n d s i p h o n
t r a n s i t i o n s . T r a n s . A S C E , 92:1435,
4.
I s r a e l s e n , O r s o n W. 1953. I r r i g a t i o n principles a n d p r a c t i c e s
J o h n Wiley & S o n s , I n c . , N@w Y o r k , New York. p. 43-51,
5.
K i n g , H o r a c e W i l l i a m s . 1954. Handbook of h y d r a u l i c s . M c G r a w H i l l Book C o m p a n y , I n c . , New Y o r k , New Y o r k . p. 9 , 3 4 - 3 7 .
6.
L u d w i g , J . H . and R . C . Ludwig. 1951. D e s i g n of PalmerB o w l u s f l u m e s . S e w a g e a n d I n d u s t r i a l W a s t e s , 23(9):1096-1107.
7.
M u r p h y , G. 1950. S i m i l i t u d e in E n g i n e e r i n g .
Company, New Y o r k , New Y o r k . p. 137-184.
8.
P a l m e r , H. K . , a n d F. D . B o w l u s . 1936. A d a p t a t i o n of ventLiri
f l u m e s t o flow m e a s u r e m e n t s in c o n d u i t s . T r a n s . A S C E ,
101:1195-1216. D i s c u s s i o n b y F. A r r e d i , p. 1231-1235.
9.
P a r s h a l l , R . L. 1945. I m p r o v i n g t h e d i s t r i b u t i o n of w a t e r t o
f a r m e r s b y u s e of t h e P a r s h a l l f l u m e . SCS B u l l e t i n 488, U. S .
D e p a r t m e n t of A g r i c u l t u r e . M a y .
10.
P a r s h a l l , R . L. 1950. M e a s u r i n g w a t e r in i r r i g a t i o n c h a n n e l s
w i t h P a r s h a l l f l u m e s and small w e i r s . SCS C i r c u l a r No. 8 4 3 ,
U. S. D e p a r t m e n t of A g r i c u l t u r e . M a y .
11.
P a r s h a l l , R . L . 1953. P a r s h a l l f l u m e s of l a r g e s i z e . SCS
B u l l e t i n 426-A, U . S . D e p a r t m e n t of A g r i c u l t u r e . M a r c h .
12.
R o b i n s o n , A . R . 1961. Study of t h e B e a v e r C r e e k m e a s u r i n g
f l u m e s . R e p o r t CER61ARR-10, Civil Engineering Section.
Colorado State University. F e b r u a r y .
The Ronald Press
13.
Robinson, A . R . 1964. W a t e r m e a s u r e m e n t s in s m a l l i r r i g a t i o n
c h a n n e l s u s i n g t r a p e z o i d a l f l u m e s . P a p e r No. 64-210, p r e s e n t e d
a t 1964 a n n u a l m e e t i n g of ASAE. F o r t C o l l i n s , C o l o r a d o . J u n e .
14.
R o b i n s o n , A . R . , and A . R. C h a m b e r l a i n . 1960. T r a p e z o i d a l
f l u m e s f o r open c h a n n e l f l o w m e a s u r e m e n t . T r a n s . ASAE
3(2):120- 128.
15.
R o b i n s o n , A . R . , a n d A . R . C h a m b e r l a i n . 1962. F l o w
m e a s u r e m e n t i n open c h a n n e l s . R e p o r t CER62ARR-ARC4.
Colorado State University. February.
16.
S e d o v , L. I. 1959. S i m i l a r i t y a n d D i m e n s i o n a l M e t h o d in
M e c h a n i c s , A c a d e m i c P r e s s . New Y o r k , New Y o r k . p. 1-24, 27-28,
17.
S k o g e r b o e , G. V . , a n d V. E . H a n s e n . 1964. C a l i b r a t i o n of
irrigation headgates by model analysis. Engineering Experiment
S t a t i o n . U t a h S t a t e U n i v e r s i t y . L o g a n , Utah. M a r c h . p. 11-15.
18.
S k o g e r b o e , G . V . , a n d W. R. W a l k e r , and L . R . R o b i n s o n . 1965.
D e s i g n , o p e r a t i o n , a n d c a l i b r a t i o n of t h e C a n a l "A" s u b m e r g e d
r e c t a n g u l a r m e a s u r i n g f l u m e . R e p o r t P R - W G 2 4 - 3 , Utah W a t e r
R e s e a r c h L a b o r a t o r y , Utah State University. M a r c h .
19.
U. S. B u r e a u of R e c l a m a t i o n , D e p a r t m e n t of t h e I n t e r i o r . 1953.
W a t e r m e a s u r e m e n t m a n u a l . U. S. G o v e r n m e n t P r i n t i n g Office.
p. 2 9 - 7 0 , 189-206.
20.
W e l l s , E . A . , a n d H . B . G o t a a s . 1948. D e s i g n of v e n t u r i f l u m e s
863312-400.
i n c i r c u l a r c o n d u i t s . P r o c . A S C E , J . S a n . E n g . Div.
P a p e r N o . 938.
T a b l e 1. B a s i c m e a s u r e m e n t s .
-
Run
No.
am
30
2. 31
0 . 693
0.562
0.452
super.
31
32
33
34
35
2.
2.
2.
2.
2.
31
31
31
31
31
0. 6 9 3
0. 6 9 4
0.708
0.733
0. 784
0.521
0.599
0.626
0.678
0.744
0.437
0.490
0. 568
0. 637
0. 722
36
37
38
39
40
2.31
2.75
2.75
2.75
2.75
0.847
0. 764
0. 762
0. 764
0. 780
0.821
0.653
0.637
0.665
0, 694
41
42
43
44
45
2.75
2.75
2.02
2.02
2.02
0. 817
0.843
0. 649
0. 649
0.650
46
47
48
49
50
2.02
2.02
2. 02
1. 78
1. 78
0.665
0.691
0. 732
0.603
0.603
(hl)m
(h )
m m
Type
(hl)p
(h4)p
(h )
m P
300. 0
4.851
3.934
3. 164
super.
crit.
sub.
sub.
sub.
300. 0
300. 0
300.0
300.0
300.0
4.851
4.858
4.956
5.131
5.488
3.647
4. 1 9 3
4.382
4 . 746
5.208
3.059
3.430
3.976
4.459
5. 054
0.815
0.540
0.516
0.549
0. 628
sub.
super.
super.
crit.
sub.
300.0
357.1
357.1
357.1
357.1
5.929
5.348
5.334
5.348
5.460
5. 747
4.571
4.459
4.655
4.858
5. 705
3. 780
3.612
3. 8 4 3
4 . 396
0.757
0. 796
0.522
0.483
0.554
0. 721
0.768
0.422
0.412
0.456
sub.
sub.
super.
super.
crit.
357.1
357.1
262. 3
262.3
262.3
5 . 719
5.901
4.543
4.543
4.550
5.299
5.572
3.654
3.381
3.878
5.047
5.376
2.954
2. 896
3. 192
0.584
0.633
0. 692
0.480
0.438
0.528
0.597
0. 670
0 . 389
0.380
sub.
sub.
sub.
super.
super.
262. 3
262. 3
262.3
231.1
231.1
4.655
4.837
5.124
4.221
4.221
4.088
4.431
4.844
3.360
3.066
3.
4.
4.
2.
2.
(h4)m
0f.
Flow
QP
696
179
690
723
660
-4
J
r.
T a b l e 1. Continued
Run
No.
Qm
(hl'm
(h4'rn
( hm )m
Type
of
Flow
Q
P
(hl$
(h4)p
(h )
m P
181.8
181.8
155.8
155.8
155.8
4.291
5. 600
3.395
3.395
3.402
4.088
5.530
2 . 506
2.226
2.695
3.934
5.425
2.114
2.079
2.394
crit.
sub.
sub.
sub.
super.
super.
crit.
sub.
sub.
sub.
super.
super.
crit.
sub.
sub.
66
67
68
69
70
1 . 40
1.40
1.20
1.20
1.20
0.613
0.800
0.485
0.485
0.486
0.584
0. 790
0.364
0.318
0.385
0 . 562
0. 775
0. 302
0.297
0.342
sub.
sub.
super.
super.
crit.
T a b l e 1.
Run
No.
Continued
Q
m
(h1)m
(h4)m
(hm)m
Type
of
Flow
sub.
sub.
sub.
sub.
super.
super.
crit.
sub.
sub.
sub.
sub.
super.
super.
super.
super.
sub.
sub.
sub.
super.
Q
P
(hl)p
(h4'p
(h )
m P
T a b l e 1. C o n t i n u e d
Run
No.
Q
m
(h 1)m
(h4)m
(hm)m
Type
of
Flow
Q
P
(hl'p
(h4lp
( hm ) P
91
92
93
94
95
0.580
0.580
0.580
0.580
0.580
0.321
0.322
0.324
0. 327
0.344
0.205
0.235
0.250
0.263
0.308
0.212
0.221
0.227
0.239
0.283
super.
super.
super.
sub.
sub.
75.3
75.3
75.3
75.3
75.3
2.247
2.254
2.268
2.289
2.408
1.435
1.645
1. 750
1.841
2.156
1.484
1.547
1. 589
1.673
1.981
96
97
98
99
100
0.580
0.407
0.407
0.407
0.407
0.589
0.266
0.266
0.266
0.267
0.588
0. 1 9 5
0.168
0.202
0.217
0.581
0. 1 7 3
0.167
0. 1 7 8
0. 1 9 3
sub.
super.
super.
crit.
sub.
75.3
52.9
52.9
52.9
52.9
4. 1 2 3
1, 862
1. 862
1.862
1.869
4.116
1.365
1.176
1.414
1.519
4 . 067
1.211
1. 169
1.246
1 . 351
101
102
103
104
105
0.407
0.407
0.215
0.215
0.215
0.283
0.678
0.181
0. 1 8 1
0.181
0.251
0.678
0. 128
0.118
0. 1 3 4
0.228
0.678
0. 119
0. 1 1 8
0.120
sub.
sub.
super.
super.
super.
52.9
52.9
27.9
27.9
27,9
1.981
4 . 746
1.267
1.267
1.267
1.757
4.746
0,896
0.826
0.938
1. 596
4. 746
0.833
0 . 826
0.840
106
107
108
109
110
0.215
0.215
0.215
0.215
0.215
0.183
0.183
0. 189
0.215
0.270
0. 1 4 3
0. 146
0. 166
0.204
0.266
0.122
0. 1 2 6
0.146
0.187
0.262
super.
super.
sub.
sub.
sub.
27. 9
27. 9
27.9
27. 9
27. 9
1.281
1.281
1.323
1.505
1.890
1.001
1.022
1. 162
1.428
1.862
0.854
0.882
1.022
1. 309
1.834
111
0.215
0.407
0.407
0.408
sub.
27. 9
2.849
2.849
2.856
4
Ln
T a b l e 2. C o m p u t a t i o n o f p a r a m e t e r s .
Run
No.
Q
P
C
C'
F
1
F
max
h4'hl
(hl- h 4 ) p
(h
)
E P
b l - h4)
h
m
-
Run
No.
Q
C
P
C'
1
F
max
h4'hl
(hl-h4)p
(h )
m P
-
(h1-h4)
-
1
F
max
Run
No.
Q
P
C
C'
1
F
max
h4'hl
(hl-h4)p
(h )
mP
(hl-h*)
h
m
-
e q e a pue r u e x 8 o . I ~~ a ~ n d u r o
MoIJ
3
pafj~awqns
g x:puaddy
The computer p r o g r a m a s listed below gives the solution to:
PRAM
m e a s u r e d value of e n e r g y l o s s p a r a m e t e r .
PRAME
value of energy l o s s p a r a m e t e r a s computed f r o m
Equation 24.
SUB
m e a s u r e d value of s u b m e r g e n c e .
SUBE
value of submergence a s computed f r o m Equation 2 3 .
QESUB
value of d i s c h a r g e a s computed f r o m Equation 28.
DIF S B
difference in m e a s u r e d d i s c h a r g e and value
computed f r o m Equation 28.
QEFR
value of d i s c h a r g e a s computed f r o m Equation 2 7
DIFFR
difference in m e a s u r e d d i s c h a r g e and value
computed f r o m Equation 2 7 .
COMPUTATION O F PARAMETERS
CALCULATION O F F L O W R A T E USING S U B M E R G E N C E F O R A T R A P E Z O I D A L F L U M E
DIMENSION H ( 4 ) , B ( 2 ) , A ( 2 ) , T ( 2 ) , F ( 2 )
H R E P R E S E N T S W A T E R D E P T H , B = F L U M E B O T T O M WIDTH, A = A R E A , T = W A T E R
S U R F A C E WIDTH
F = F R O U D E N U M B E R , Q = F L O W R A T E , P O I N T 1 = E N T R A N C E , 2 = P O I N T O F MIN.
DEPTH, 4 = EXIT
S L O P E O F F L U M E SIDE WAS A ONE T O ONE
1 READ 100, B ( l ) , B ( 2 )
100 F O R M A T ( 2 F 6 . 2 )
P U N C H 200
200 F O R M A T (1X5HDIFFR3X4HQEFR5X1HQ5X5HQESUB4X5HDIFSB4X3HSUB4X4HSUBE4X
14HPRAM4X5HPRAMEl/ )
2 READ 101, Q , H(1), H(Z), H(4)
101 F O R M A T ( F 6 . 2 , 3 F 8 . 5 )
G = 32.2
A (1) = (B(1) t H(l))*H(l)
A ( 2 ) = ( B ( 2 ) t H(2))':'H(2)
Z = SQR ( A ( I ) / A ( Z ) * A ( l ) / A ( 2 )- 1. 0 )
3 Q T = A(l)::'SQR(G*(H(l) - H ( 2 ) ) ) l Z
C
Q T IS T H E T H E O R E T I C A L F L O W R A T E
C = Q/QT
C IS A DIMENSIONLESS NUMBER AND A F O R M O F A F R O U D E NUMBER
C
cc = c / z
T ( 1 ) = B ( 1 ) t 2.0>%H(1)
T ( 2 ) = B ( 2 ) t 2. O*H(Z)
4 F(1) = Q/(A(l)::'SQR(G*A(l)/T(l)))
5 F ( 2 ) = Q/(A(2)::'SQR(G::'A(2)/T(2)))
6 SUB = H ( 4 ) / H ( 1 )
C
C
C
C
C
C
C
COMPUTATION O F P A R A M E T E R S ( c o n t i n u e d )
C
SUB IS T H E D E G R E E T H E F L O W IS S U B M E R G E D
PRAM = (H(1) - H ( 4 ) ) / H ( 2 )
C
S T A T E M E N T S 1 0 , 1 1 , 1 2 , 13 A R E E M P I R I C A L L Y D E R I V E D EQUATIONS
34*PRAM
10 S U B E = 0 . 9 9 1 10. 0::O.
1 1 P R A M E = 0. 300:?F(2):%*:2.38
12 Q E F R = 34. 7:KF(2)*H(2):?:%1.74
1 3 QESUB = - 13. 83"(H(1) - H(4))*;*1. 7 4 / ( L O G ( H ( 4 ) / . 99::H(1))2;.43429):%:::1. 32
20 D I F F R = Q - Q E F R
2 1 DIFSB = Q - QESUB
P U N C H 2 0 1 , D I F F R , Q E F R , Q , QESUB, D I F S B , S U B , S U B E , P R A M , P R A M E
201 F O R M A T ( F 6 . 3 , 3 F 8 . 2 , F 8 . 3 , 4 F 8 . 4 )
IF (SENSE SWITCH 9 ) 3 0 , 2
30 P A U S E
END
16. 0
6. 0
6P8 ' Z
298 ' I
8ZP.1
Z91 ' 1
9PL ' P
L5L ' I
615 ' 1
911 ' P
951 ' Z
IP8.1
£96 'P
6£9 ' Z
962 ' Z
I0Z ' 5
825 '£
286 ' Z
L65 ' Z
ZL5 ' 5
LZ6 'E
9 0 2 .£
0P6 ' Z
025 ' 5
680 'P
PPP "2
b 6 0 'E
L55 '77
P6L 'E
6££ '£
68£ ' P
906 '£
865 '£
PP8 '.6
I£=P.P
RRO ' P
ZLE ' 5
66Z ' 5
6 5 s ';b
LPL ' 5
80Z ' 5
9PL ' P
Z8£ ' P
058 ' Z
068 'I
505 ' 1
£ZE ' 1
L%L ' P
186'1
698 ' 1
FZ1 '=P
80P ' Z
682 ' Z
7786 '-6
506 ' Z
L£L ' 2
9£Z ' 5
9 6 9 ~£
F8Z 'E
621 ' £
1Z9 ' 5
560 ' P
LL5 ' £
ZLP 'E
009 ' 5
16Z '.6
P98 'E
5PL * £
9.6L ' P
002 'P
8 1 0 '.6
£89 'P
96s
S9Z ' P
.6ZT .5
LC8 ' P
5 5 9 .P
1 0 6 .5
61L ' 5
09P ' 5
6Z6 ' 5
88P ' 5
1E1 .5
956 ' 9
'*
6
6
6
6
6
6
6
£
£
£
9
9
9
'LZO
'LZ0
'LZ0
'LZ0
'Z50
'250
'250
'5L0
'5L0
'5L0
'Z01
'ZOI
'Z01
z'1f1
Z'1E1
Z'IET
Z '1£1
8 '551
8 '551
8 '551
8 '551
8'181
8 '187
8 '181
8 '181
8 'LOZ
8 'LOZ
8 'LOZ
1 .1£Z
1 'T£Z
7 '1SZ
E 'Z9Z
£ 'Z9Z
£ 'Z9Z
1 'L5£
1 'L5S
I 'L5F
0 '00s
0 '00£
0 '00£
0 'OOE
Table 4.
DIFFR
Continued.
QEFR
Q
DIFSB
QESUB
SUB
SUBE
PRAM
PRAME
%ERROR
(DIFSBIQ)