measuring open channel flows with compound weirs

1. The 11th International Symposium on ADVANCED TOPICS IN ELECTRICAL ENGINEERING 2019
March 28-30, 2019, Bucharest, Romania
Introduction
Under climate change conditions, water management in small, headwater
catchments, becomes a must, mainly for small communities and farmers.
Regrettably, headwater catchments are not statistically monitored at gauging
stations in terms of discharge. To overcome this deficiency, different types of
structures for measuring the discharge in small streams and irrigation canals
have to be set in place. Of these, weirs are the most common, since they
are reliable, very simple, have good accuracy, low cost and are easy to
install, operate and maintain. With proper selection of the shape and
dimensions of the overflow section, and following the weir calibration for all
ranges of operating conditions, stage – discharge relationship can be
developed and used for practical purposes. In this context, the paper
objective is two fold:
• To propose a simple method to calibrate a typical triangular-rectangular
compound weir in laboratory, that could be used in practice for locations
subjected to frequent and important flowrate variations, triggered by
climate change conditions, and ranging from low discharges during
drought periods to large discharges during heavy rainfalls or storms;
• To propose a method of developing a stage – discharge relationship
for a real scale measuring device, based on measurements performed on
small scale laboratory models, using the Froude similitude criterion.
Experimental setup and method
Results
Actual flow rate as a function of measured total head over the compound
weir is presented in Fig. 4.
Conclusions
Calibration of compound weirs must be made in stages, since a discontinuity
in the variation of the discharge coefficient with head appears in the
transition stage, where the opening changes shape and the flow area
increases very much.
For the considered triangular-rectangular compound weir, the best
regression functions for the discharge coefficient were found to be power law
for the V-notch and a 3rd degree polynomial for the rectangular – upper part.
The two branches of the head-discharge curve of the compound weir were
fitted independently with power law functions with good mean square error
ratio values of 0.9707 for the lower, V-notch, stage and 0.9968, for the upper
stage. However, the overall head discharge curve for the compound weir
was found to be best fitted by a 3rd degree polynomial regression function
(with a mean square error of R2 = 0.999).
Considering the same acceleration of gravity and Froude similitude, a first
approximation of the real-scale rating curve of a prototype can be obtained
from the rating curve of the laboratory model.
The prototype should further be corrected by recalibrating on-site with
specific field discharge measurement techniques. Since the sensitivity of the
V-notch is better than the rectangular weir, the lower stage is used for
measuring small discharges, while the upper stage is used for higher
discharge values, during intense rainfall events.
Fig.1 Rectangular and trianghiular weir Fig. 2 Compound weir
b1/2
h2
h1
b
p
B
3
2
1
2
3
h1
p
B
weir
b
p
h2
B

Theoretical background
800
100
400
2000
2000
500
2000 12000 800
Fig.3 . Experimental setup: lateral and plane view
y = 0.8672x2.5042
R² = 0.9707
y = 1.115x2.5942
R² = 0.9968
0.000
0.001
0.002
0.003
0.004
0 0.05 0.1 0.15
Q
(m
3
/s)
h (m)
V-notch
compound
Dependance of the global discharge coefficient from eq. (5) of the
compound weir on the non-dimensional head/weir wall ratio, Cd = f(h/p)
was computed and plotted in Fig. 5. Two different functions were sought
to best approximate the two branches: a power law for the V-notch and a
3rd order polynomial for the rectangular notch.
Both stages of the head discharge curve for the compound weir, were
found to be best fitted by a 3rd degree order polynomial regression
function (Fig. 6).
In Fig. 7 and Fig. 8 visualisations of the nappe are shown for two different
head values.
Fig. 4 Discharge-head curve of the compound weir
y = 0.3029x-0.806
R² = 0.9798 y = -2.2725x3 + 1.3804x2
+ 0.5315x + 0.2886
R² = 0.8976
0.54
0.56
0.58
0.60
0.62
0.64
0.66
0.68
0.70
0.3 0.4 0.5 0.6 0.7
Cd
h/p
V-notch
rectangular weir
Fig. 5 . Cd = f(h/p) fitted with two different functions
h = - 9.8124Q3 + 3.1181Q2
- 0.2567Q + 0.0072
R² = 0.999
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.000 0.002 0.004 0.006
h
(m)
Q (m3/s)
V-notch
compound weir
Fig. 6 . Head-discharge curve of the compound weir
0
5
10
15
20
25
30
35
0.000 0.002 0.004
dh/dQ
Q (m3/s)
V-notch
compound weir
Fig. 7 . Flow nappe over the compound weir around the transition from
V-notch to rectangular opening
Fig. 8 . Flow nappe over the compound weir for a higher head, above
the transition head from V-notch to rectangular opening
Fig.9. Sensitivity change in weir
head for unit change in discharge
as a function of discharge
Calibration of a compound weir for discharge
measurement in small headwater streams
subjected to climate change
Daniela-Elena GOGOAȘE NISTORAN, Cristina-Sorana IONESCU, Ioana OPRIȘ, Sorina COSTINAȘ
University POLITEHNICA of Bucharest, Faculty of Power Engineering
daniela.nistoran@upb.ro, cristina.ionescu@upb.ro, ioana.opris@upb.ro, sorina.costinas@upb.ro
The general head-discharge relationship for a simple opening weir (Fig. 1) is
a power law
(1)
where K is a dimensional coefficient, Cd - the non-dimensional discharge
coefficient, and n - the exponent of the weir head depending on the opening
shape.
Cd takes into account friction, surface tension, changes in pressure
distribution due to streamline curvature, nappe lateral contraction, its vertical
drawdown and velocity profile/distribution in the approach section.
Dimensional analysis provides for Cd the following functional relationship
(2)
where p is the weir wall height (bottom to crest/vertex), B is the width of the
approach channel, Re - the Reynolds number, and We – the Weber
number.
The compound weir (Fig. 2) considered in the present paper consists of a
rectangular notch over a V-notch. It provides an optimal method to
accurately measure small flowrates while also having the capacity to
measure large ones without too much increase of the water head.
n
d t
Q Kh C Q
 
, , ,Re, We, geometry
d
h h p
C f
p B B
 
  
 
Neglecting the flow contraction, the theoretical discharge through the
compound weir may be derived by integrating the flow velocity over a
horizontal slice of elementary area, obtaining:
(3)
To compute the actual flow, the entire flow area was assimilated with a large
V-notch (1) plus a rectangular area of b width (2) minus a small V-notch
overlapped area [14] (Fig. 2) and two distinct discharge coefficients were
taken into account to correct the theoretical formula, one for both V-notches,
Cd1 and one for the rectangular weir, Cd2, as follows:
(4)
A global discharge coefficient for the entire compound weir might be defined
as:
. (5).
Cd includes both the V-notch and the rectangular weirs coefficients, which
were calibrated in order for the compound weir to measure accurately the
discharge.
 
5/2 5/2 3 2
. . 1 2 2 2
8 2
2 2
15 2 3
comp th
Q g tg h h h g b h

   
      
   
 
 
 
5/2 5/2
1 1 2 2
3 2
2 2 2
rectaangular weir
8
2
15 2
2
2 0,2
3
compound d
d
V notch
Q C g tg h h h
C g b h h
Q Q


   
   
   
 
     
 
.
actual
d
compound th
Q
C
Q

The experimental setup consists of a horizontal 12m long glass-walled
recirculating flume with rectangular cross-section of 0.4 m in width and 0.5 m
in depth (Fig. 3). The compound sharp-crested weir was built of thin 2mm
aluminium plates and screwed to a frame sealed on the bottom and walls of
the flume at a distance of 5m from the flume upstream end and 7m from the
flume downstream end (Fig. 3).
The geometry of the compound weir (Fig. 4) was designed so that it would
cover the whole range of discharges flowing through the existing flume p =
18cm, b = 12.4cm, 2l = 3.2 cm., h1 max = 8cm and h2 max = 24cm..
Water was supplied to the flume from an overhead tank at constant head.
Recirculation was provided by a centrifugal pump and a pipeline fitted with a
calibrated diaphragm and valves to control the flow rate. The actual flow rate
was measured with a calibrated diaphragm. The water depth was measured
by means of a vernier point gauge level indicator with a accuracy of 0.2 mm.
The two branches of the rating curve of the compound weir were fitted
independently with power law functions with very good mean square ratio
values (Fig. 4). One may notice the lower stage power law function has
an exponent very close to 2.5 (characteristic for the V-notch), whereas the
exponent of the upper stage has a different value.
The average value of the two discharge coeffcients was found to be from
curve fitting (Fig. 4) and equation (4): Cd1 = 0.6538 and then Cd2 = 0.622.
The analisys of the sensitivity dh/dQ was obtained by differentiating Q with
respect to h. Using the power law function of Q with h from Fig. 4 two
equations were obtained:
(6)
and
(7)
Sensitivity change as a
function of discharge
presented in Fig. 9.
It shows the lower stage V-
notch has a higher sensitivity
than the upper stage
compound weir.
1.5942
/ 0.4605
dh dQ h
 
1.5042
/ 0.3457
dh dQ h
 