An Experimental Study for Radial-Gated Ogee Spillway Discharge and
Comparison with Other Model Studies
TEFARUK HAKTANIR, EROL BOR
Department of Civil Engineering,
Nuh Naci Yazgan University,
Erkilet Dere Mahallesi, Kayseri,
TURKEY
Abstract: - The book "Design of Small Dams" is a renowned reference publication used for the design of dams
in America and it is commonly used in Turkey as well. Flood spillways of many dams are ogee profile radial-
gated weirs, and accurate calculation of discharges passing over a partially open radial-gated ogee spillway is
important in flood routing computations. Because each spillway and its appurtenant structures have geometrical
and hydraulic properties peculiar to their own, a generalizable method for calculating the discharge over a
partially open radial-gated spillway should not be realistic. Therefore, many laboratory experimental studies are
done both in America and in Turkey for more accurate calculation of spillway discharge of dams individually.
With the objective of comparing the results of these studies and the method given in that book another
experimental study is performed as summarized in this paper. For this purpose, experiments for the partially
open radial gate in a laboratory setup having a 95 mm high, 10 cm wide ogee spillway model with an adjustable
radial gate of proportionate dimensions both placed in a 5 meter long, 10 cm wide and 30 cm high channel are
done with many combinations of flow rates and gate openings as allowed by the physical dimensions of the
open channel setup available in our laboratory and the maximum flow rate its pump can generate. Taking a
scale ratio of 1/100, the relative differences of the discharge coefficients given in the above-mentioned book
from the experimental discharge coefficients measured in this study for the same configurations are found to be
between +3% and +34%. Although the discharge coefficients determined by this experimental study are not too
deviant from those obtained from 15 laboratory model studies done by the United States Bureau of Reclamation
and the United States Army Corps of Engineers in America and seven such studies done by the State Water
Works in Turkey, it is concluded that the discharge – head relationship of a radial-gated spillway of a dam
should be determined by a laboratory experiment on a model of not too small a scale, and a generalized method
cannot be applicable to all spillways in the world as a whole.
Key-Words: - Radial-gated ogee spillways, discharge over partially open radial-gated ogee spillways, discharge
coefficients of radial-gated ogee spillways, crest profiles of ogee spillways, radial-gated ogee
spillway model, Froude similarity between model and prototype, open channel flows.
Received: March 7, 2024. Revised: October 3, 2024. Accepted: November 6, 2024. Published: December 2, 2024.
1 Introduction
Flood spillways of almost all dams in Turkey have
ogee profiles described in [1], [2] and a majority of
them are equipped with radial gates [3], [4], [5].
Because it is an orifice flow under a gate of circular
arc rather than a planar sluice, flow under a partially
open radial gate over an ogee spillway is more
complicated than the fully open (free flow) case. In
spite of this fact, the former needs only one chart for
the discharge coefficient, while the latter has three
charts, [1]. An explanation of the method for the
fully open case with numerical examples is given on
seven pages, while the explanation for the partially
open case is given on one page only, where there is
a figure depicting the discharge coefficient. This
figure in [1] is a replica of the same figure in [6]. In
this figure, there are widespread noises of the
plotted points around the fitted curve for the case
where the gate seat is a little downstream from the
spillway apex, which was drawn using the data
measured only on two laboratory model spillways
and three spillways of actual dams, [6]. The curve
for the case where the gate seat is at the apex is
drawn based on only one laboratory model data [6].
The equation given both in [1] and in [6] for the
discharge passing through a partially open radial
gate over an ogee spillway, which is a
dimensionally homogeneous equation, is:
(1)
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where, C is the discharge coefficient, D is the
shortest distance between the gate lip and the
spillway crest curve, L is the net length of the
spillway crest, and H is the vertical difference
between the total head just upstream of the gate
(including the velocity head of approach) and the
center of the gate opening. The relevant figure in [1]
gives C as a function of the angle (Ɵ) between the
tangent to the gate lip and the tangent to the crest
curve at the point closest to the gate lip.
Computation of this angle Ɵ and of D is fairly
complicated and requires involved trigonometric
and geometric analyses. A long numerical and
analytical algorithm is suggested in [6] for
computations of both Ɵ and D, which requires (1)
manually plotting both the crest coordinates and the
slope function of the analytical ogee profile in a log-
log scale millimetric graph paper and visually
reading the numerical values out of this chart, (2)
replacing the analytical expression of the
downstream part of the ogee profile by arcs of three
circles of different center points and radius lengths,
(3) doing computations on a numerical table
comprising 20 columns for the gate opening D and
of the angle Ɵ, and (4) doing computations on
another table having 15 columns for ultimately
determining the discharge Q, [6]. This manual
method, which is advocated by [1] also, presents an
archaic approach as if we were in 1950s and 1960s
and it will definitely take a long time and a large
effort improper to the current age of computers.
Another deficiency of this method is that it does not
present any algorithm for a possible case where the
closest point between a partially open gate lip and
the spillway crest lies on the part of the crest curve
upstream from the spillway apex which is part of a
circle and not the curve depicted by the analytical
expression defining the downstream face of an ogee
spillway. A real-life example to such a case is the
radial gate of Yellowtail after Bay Dam in Montana
USA when the gate opening is 6 ft, [7].
There should exist many ogee spillways
equipped with radial gates in the world, and
controlled releases of floods of moderate
magnitudes through partially open radial gates
should be a common practice. Hence, accurate
computation of discharges under partially open
radial gates is an important problem. There are two
issues about the computation of discharge passing
under a partially open radial gate over an ogee
spillway. First, the curve proposed in [1] is not
sufficient for a general application because it is
developed using the measurements on three actual-
size spillways and two laboratory models only.
Besides, even in its present form, the observed
points have large noises around the fitted curve,
especially for values of angle Ɵ smaller than 73º,
[6]. Secondly, the method advocated in [1] for
computing the flow rate is too cumbersome,
intricate, and time-consuming as explained in the
previous paragraph. To remediate the old-fashioned
and winding method of computation for Q in
equation 1, a novel analytical and numerical
algorithm was proposed in [7], and unfortunately, it
has gone unnoticed so far although it presented a
modern method which is executed in a split second
in an ordinary computer requiring neither initially
prepared manually-drawn graphs nor numerical
tables with many columns, [7].
To probe into the apparent problem of doubt for
the relevant figure in [1] about its being a general
chart applicable to any spillway in the world, a
study was performed whose details and results are in
[8]. In that study, using data from 6 reports of
laboratory model studies done by the United States
Bureau of Reclamation, 9 such reports done by the
United States Army Corps of Engineers, 6 such
reports, and one report of measurements directly on
the flood spillway of Seyhan Dam done by the
Technical Research and Quality Control Department
of the State Water Works of Turkey, all having
radial-gated ogee spillways, the experimentally
determined discharge coefficients for partially open
gates were compared with those given by the
method of [1], whose results revealed large
differences between the two and there did not seem
to have a generalizable chart. In that study it is
concluded that a general chart is not possible, and
the best thing to do is to obtain the chart for the
discharge coefficient peculiar to each spillway by
doing laboratory measurements of all relevant
quantities on not too small a model [8]. It is further
recommended that measurements in the approach
channel of actual spillways during days of incoming
floods should be taken because recent technology
would allow such measurements at reasonable costs
with no danger to life, [8].
Recently, we have had an open channel setup
built by a professional laboratory equipment
manufacturing company with dimensions of 5 m
length, 10 cm width, and 30 cm wall height. The
channel is rectangular and it has a self-circulating
water flow provided by a bottom storage tank and a
pump of suitable capacities. On the pipe supplying
the flow to the upstream entrance of the channel,
there is a high-precision flow meter for measuring
the circulating discharge. Figure 1 shows the
photograph of this open channel model. Recently,
we did experiments for a partially open radial gate
in this laboratory setup on an ogee-profile spillway
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model having a height of 95 mm and a width of 10
cm and an adjustable radial gate of proportionate
dimensions mounted to the top of the side walls
above the spillway with the help of screws placed in
the middle of the channel with many combinations
of flow rates and gate openings, [9]. The maximum
model flow rate was chosen so that the upstream
water depth would not exceed the wall height of the
channel, and the minimum discharge was
determined so that the spillway flow would not be
free flow with a gate opening of 1 cm. By trials, we
have determined the maximum radial gate opening
as 4.5 cm so that with that opening and the
maximum flow rate the upstream water depth would
not overflow the channel wall. We decided that
increments of 0.5 cm for radial gate openings were
sufficient for our experiments. We had two
objectives in our study: (1) to observe how close the
discharge coefficients to be measured on this
laboratory model would be to the ones observed in
those 22 detailed laboratory studies done by
prestigious organizations in America and in Turkey,
and (2) to check the agreement of the generalized
method for the discharge coefficient for the case of
partially open radial-gated ogee spillways given in
[1] with the ones observed experimentally on those
laboratory studies.
2 Material and Method
2.1 Data of 21 Models and One Actual
Spillway of Previous Studies and of the
Model of this Study
All relevant geometrical and hydraulic data of 21
laboratory model spillways and one actual spillway
presented in the technical reports cited in [8] are the
initial part of the material used in this study. We
obtained permissions from these three organizations
for the usage of the data in those reports. The
second material is the data obtained from the
experiments done with 29 combinations of flow
rates and gate openings in the laboratory setup of
this study. The ranges of flow rates and gate
openings were constrained by two criteria: (1) the
upstream flow depth would not exceed the wall
height of the channel and (2) the gate opening
would be lower than the free flow formation. The
flow rates generated in the setup were 150, 130,
110, 90, 70, 50 liters/minute, and the vertical gate
openings were between 1.0 cm and 4.5 cm at 0.5 cm
increments, [9]. Measurements of the relevant
quantities were done in a few minutes after each
setting when the flow conditions and the overall
water surface profile became steady. Figure 2 shows
the instant of one of the experiments. As seen in
Figure 2, the radial gate is free to move up and
down with the help of a steel rod. The vertical
distance between the lower tip of the gate and the
apex of the spillway and the flow depths were
measured by a millimetric depth measurement
gauge, [9].
The minimum flow rate and the minimum gate
opening generated in the experiments were
0.0008333 m3/s (= 50 lt/min) and 0.01 m. Because
the kinematic viscosity of water is: ν ≈ 110–6 m2/s,
the Reynold Number of the flow passing over the 10
cm wide spillway under a 1 cm opening is: Re =
8333. The Re magnitudes for all other flows in the
channel and under the gate are greater than 8333
and about 25,000. Therefore, all the flows in all
conditions of our experiments were turbulent (Re >
2000).
Because the widths of both the spillway and the
inner side of the rectangular channel were 0.1 m, we
assumed the scale of the model to be 1/100. Hence,
ours was a single gate, single opening spillway
corresponding to a prototype dimension of 10 m.
Applying the Froude similarity principle to this
setup with a length ratio of Lp/Lm = 100, the
relationship between the prototype and the model
flow rates becomes:
(2)
Here, the subscripts p and m denote “prototype”
and “model”. The maximum flow rate we could
obtain in this setup without flowing over the side
walls was 150 liters/minute, and this corresponds to
a prototype flow rate of 250 m3/s. The steady
conditions for a combination of flow rate and gate
opening were reached in a few minutes, and the
circulating discharge was read from the built-in flow
meter in liters/minute. Altogether 29 combinations
of flow rates and gate openings were configured
during the experiments.
Fig. 1: Photograph of the open channel model setup
used in the experiments
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Fig. 2: Instant of one of the experiments of flow
passing under the partially open radial gate.
2.2 n and K Coefficients of the Downstream
Crest Curve of the Model Spillway
The analytical expression of the crest curve of the
downstream face of an ogee spillway is [1]:
(3)
where, y and x are the ordinate and the abscissa of
any point on the downstream profile with respect to
a coordinate system whose origin is at the apex and
the ordinate axis is downwards in m, Hd is the total
head with respect to the spillway apex for the design
discharge in m, and n and K are the coefficients. We
placed the model spillway on a large millimetric
graph paper, drew its profile on the paper, and
carefully read the abscissa and the ordinate
coordinates of both of its downstream and upstream
faces. We computed the n and K coefficients for our
model spillway by the Least-Squares method so as
to match its downstream profile to the curve defined
by equation 3 as close as possible. The sum of
squared differences between the measured ordinates
of N number of points on the downstream face and
those defined by equation 3 is:
(4)
Denoting y measuredi by yi, for the ogee profile,
SSR becomes:
(5)
The best fit n and K coefficients are those
minimizing SSR, and for that both partial
derivatives of SSR with respect to n and K must be
equal to zero. Taking ∂SSR/∂n and ∂SSR/∂K
analytically and equating them to zero, a system of
two nonlinear equations result, which in
algebraically concise forms are as given below.
(6a)
(6b)
Solution of this system by the iterative Newton-
Raphson algorithm beginning with initial estimates
taken from [1] always gives convergent cycles with
a precision of six significant digits in both roots.
The magnitudes of n and K coefficients of the ogee
spillway having a height of 9.5 m and a width of 10
m for a design discharge of 250 m3/s and Hd =
5.642 m turn out to be: n = 1.618 and K = 0.8289.
Figure 3 shows the actual downstream profile and
the one depicted by equation 3 with these
coefficients.
2.3 Experimental and Theoretical Discharge
Coefficients
The discharge coefficient which occurred for a
combination of flow rate and gate opening in our
experiments, Cexperimental, was computed by the
equation below, which is the arranged form of
equation 1.
(7)
We computed the angle between the tangent to
the gate at the gate lip and the tangent to the crest
curve at the point closest to the lip (Ɵ), the shortest
gate opening between the gate lip and the spillway
crest (D), and the theoretical discharge coefficient
(Ctheoretical) for any one of 29 combinations using
the method devised in [7].
3 Results
Table 1 (Appendix) presents the numerical results of
the experiments summarized above.
In [8] the experimentally determined discharge
coefficients are presented against the angle Ɵ for all
combinations of flow rates and gate openings
covered in the reports of those 21 laboratory models
plus one full-scale spillway mentioned in subsection
2.1 above. Together with all the coefficients in all of
those 22 reports of the previous studies, we plotted
the 29 discharge coefficients which we
experimentally determined in our laboratory setup
against the angle Ɵ all in the same figure, which is
Figure 4. This figure succinctly presents the results
of so many laboratory studies done by prestigious
organizations in America and in Turkey together
with the results of our study. Our conclusions and
discussions are given based on these results in the
ensuing sections.
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Fig. 3: The measured points and the points
computed by the analytical expression for the
downstream face of the model ogee spillway whose
n and K coefficients are determined by the Least-
Squares method
Fig. 4: Plots of the discharge coefficients against the
angle Ɵ obtained by the measured data analyzed in
[8], of the discharge coefficients obtained in [9] and
the theoretical charts given in [1]
4 Conclusions
There are two conclusions reached out of this study
which are presented below.
(1) The curves given in [1] cannot be generally
applicable. First, the curve for the case where the
spillway bottom lip is on top of the spillway apex
was obtained from one (only one) laboratory model,
not even a single actual-sized spillway. Second, the
curve for the case where the spillway seat is a little
further downstream from the spillway apex is the
best-fit curve to the measurements on three actual
spillways plus two model spillways. Besides, there
are considerably wide noises of the plotted points
about this best-fit curve for Ɵ angles smaller than
73º, which is clearly visible in the relevant chart
given in [6]. As seen in Figure 4 here, the plotted
points of the experimentally determined discharge
coefficients obtained by prestigious organizations in
America and in Turkey from laboratory model
studies exhibit very wide dispersions around a
prospective generalizable curve.
(2) When the plotted points in Figure 4 are
examined closely, it is noticed that both with our
experimental results and with those of the previous
22 studies, there are a few different discharge
coefficient (C) values against the same value of the
angle Ɵ. This clearly indicates that C depends not
only on Ɵ alone, but on another variable along with
Ɵ. For example, with our experiments, there are
three different C’s against Ɵ=56º, four different C’s
against Ɵ=59º, five different C’s against Ɵ=63º, five
different C’s against Ɵ=65º, and four different C’s
against Ɵ=68º. In a previous study, this extra
explanatory variable was found to be the ratio of the
vertical gate opening to the total head with respect
to the spillway apex, [8].
5 Discussions
As summarized heretofore, the discharge
coefficients measured in our channel model turned
out to be in the same ball park as those determined
on 21 laboratory models by prestigious
organizations in America and in Turkey and one
actual spillway in Turkey. This finding supports our
final comment that the discharge coefficients for
various combinations of flow rates and gate
openings must be determined individually based on
laboratory model studies with not too small
(model)/(prototype) length ratios because the ogee
spillway of each dam has unique peculiarities
affecting the (head)↔(gate opening)↔(flowrate)
relationship like differences in geometrical shape,
length, and roughness of the approach channel,
geometrical shapes of the approach abutments,
geometrical shapes and numbers of the piers, angle
of inclination of the upstream face of the spillway,
and position of the spillway with respect to the
embankment.
In light of all the previous experimental studies
summarized heretofore and of our experimental
study, it is obvious that the method described in [1]
for computing the discharge flowing over a partially
open radial-gated ogee spillway is not sufficient for
accurate calculation of the flow rate, and it needs to
be amended by relating the discharge coefficient to
both the angle Ɵ and to the ratio of the gate opening
to the total head with respect to the spillway apex.
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This relationship should be obtained individually for
each dam.
As seen in Figure 4, the experimentally obtained
values of the discharge coefficient (C) extend
farther away from the generalized curve (green line
in Figure 4) given in [1] both upwards and
downwards as well as sideways. This is another
indication of the lack of the method of [1] being
generally applicable.
There are as many as 40 relevant publications
cited and referenced in [8]. They are not repeated
here to save from the length of this paper.
A study of similar theme can be mentioned as
another case emphasizing the significance of such
experimental studies in determining the appropriate
discharge coefficients of spillway-type hydraulic
structures, [10].
Acknowledgement:
The experimental part of this study was performed
in the Hydraulics Laboratory of Nuh Naci Yazgan
University.
References:
[1] United States Department of the Interior
Bureau of Reclamation, Design of Small
Dams, A Water Resources Technical
Publication, Third Edition, U.S. Government
Printing Office, Washington, DC 20402-9328,
1987.
[2] United States Department of the Interior
Bureau of Reclamation, Design of Gravity
Dams, Design Manual for Gravity Dams, A
Water Resources Technical Publication, U.S.
Government Printing Office, Denver,
Colorado, 1976.
[3] Temelsu (2007) Final Design Project of the
Bayramhacılı Dam, Chapter 3.1 Hydraulic
Computations, 4. Flood Spillway. Temelsu
Engineering Services Co., Ankara, Türkiye.
[4] Şentürk F (1994) Hydraulics of Dams and
Reservoirs. Water Resources Publications, PO
Box 260026, Highlands Ranch, Colorado,
USA.
[5] International Commission on Large Dams
Turkish National Committee, Dams of Turkey,
International Commission on Large Dams
Turkish National Committee, DSİ Foundation,
2014,
https://kutuphane.tarimorman.gov.tr/vufind/R
ecord/1202282 (Accessed Date: October 20,
2024).
[6] United States Army Corps of Engineers,
Mississippi River Commission, Waterways
Experimentation Station, Hydraulic Design
Criteria, Volume 2, Tainter Gates on Spillway
Crests, Sheets 311-1 to 311-5, Department of
the Army, Corps of Engineers, Mississippi
River Commission, Waterways
Experimentation Station, Vicksburg,
Mississippi, USA, 1987.
[7] Haktanir T, Citakoglu H, Kucukgoncu H, An
efficient algorithm for ogee spillway
discharge with partially-opened radial gates
by the method of Design of Small Dams and
comparison of current and previous methods,
The International Journal of Engineering and
Science (IJES), 5(7), 2016, 13-26.
[8] Khalaf M, Comparison of Discharge
Coefficients for Partially-Opened Radial-
Gated Ogee Spillways, M. Sc. Thesis. Institute
of Technical Sciences, Erciyes Univ., Kayseri,
Türkiye, 2017, Thesis No: 467811, [Online].
https://tez.yok.gov.tr/UlusalTezMerkezi/T
ezGoster?key=q3-
d9QtLoVA2OMExHSkJpUzwb4ZGGOU
1nI0ubnC0KZ8exO-
OQSXgXXUaZVizwkVh (Accessed Date:
October 20, 2024).
[9] Bor E, An Experimental Study for Radial
Gated Spillway Discharges (In Turkish),
M.Sc. Thesis, Institute of Technical Sciences,
Nuh Naci Yazgan University, Kayseri,
Türkiye, 2023, Thesis No: 829292, [Online].
https://tez.yok.gov.tr/UlusalTezMerkezi/T
ezGoster?key=nLNfCsWgUluh5T2iyudSh
u_8lLNbINLZheEAk9BBD9Qk29_wL9K
-dmt1gbqIexuO (Accessed Date: October
20, 2024).
[10] Qasım R M, Abdulhussein I A, Al-Asadi K,
Experimental Study of Composite Inclined
Weir – Gate Hydraulic Structure, WSEAS
Transactions on Fluid Mechanics, 15, 54–61,
2020,
https://doi.org/10.37394/232013.2020.15.5.
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APPENDIX
Table 1. Discharge coefficients of equation 1 experimentally measured in this study, those theoretical ones
given by [1] for the same configurations, and the relative differences of the latter from the former (Relevant
dimensions of the corresponding prototype are presented first)
Net spillway length: 10 m, sill height of spillway: 9.5 m,
angle with vertical of the upstream face of spillway: 0º,
bottom elevation of the approach channel at the upstream face of the spillway: 100.0 m,
spillway apex elevation: 109.5 m,
spillway design head: 5.642 m,
abutment contraction coefficient: 0.0,
piers contraction coefficient: 0.0,
radius of the radial gate: 31.4 m,
radius of the circle of the spillway crest profile upstream of the apex: 2.0 m,
magnitude of n coefficient of the downstream crest curve: 1.618,
magnitude of K coefficient of the downstream crest curve: 0.8289,
elevation of the center of the gate trunnion: 114.5 m,
elevation of the gate seat: 108.5 m,
elevation of the top of the gate at closed position: 139.5 m.
Gate Flow rate Upstream flow depth Cexperimental Ctheoretical* Relative
Opening (Q) w.r.t. spillway apex (this study) difference
(m) (m3/s) (m) (%)
1.0 150 14.86 0.5887 0.6712 +14
1.0 117 10.56 0.5486 0.6712 +22
1.0 83 5.46 0.5633 0.6712 +19
1.5 217 17.28 0.6190 0.6725 +9
1.5 183 12.78 0.6149 0.6725 +9
1.5 150 9.98 0.5754 0.6725 +17
1.5 117 7.88 0.5104 0.6725 +32
1.5 83 4.58 0.5018 0.6725 +34
2.0 250 16.11 0.6095 0.6739 +11
2.0 217 12.71 0.6007 0.6739 +12
2.0 183 10.21 0.5738 0.6739 +17
2.0 150 7.51 0.5598 0.6739 +20
2.0 117 5.11 0.5506 0.6739 +22
2.5 250 12.06 0.6067 0.6753 +11
2.5 217 9.86 0.5897 0.6753 +15
2.5 183 6.86 0.6197 0.6753 +9
2.5 150 5.66 0.5730 0.6753 +18
2.5 117 4.26 0.5417 0.6753 +25
3.0 250 9.53 0.6062 0.6770 +12
3.0 217 7.93 0.5877 0.6770 +15
3.0 183 5.83 0.6076 0.6770 +11
3.0 150 4.23 0.6290 0.6770 +8
3.0 117 3.73 0.5428 0.6770 +25
3.5 250 7.60 0.6222 0.6787 +9
3.5 217 6.20 0.6191 0.6787 +10
3.5 183 4.70 0.6453 0.6787 +5
4.0 250 6.28 0.6464 0.6803 +5
4.0 217 5.18 0.6510 0.6803 +5
4.5 250 5.57 0.6600 0.6820 +3
*: C coefficients given by the relevant figure in [1] for the experimental configuration
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Tefaruk Haktanir developed the theoretical and
experimental formulation of the study. Erol Bor
performed the experiments, did the computations,
and presented the results in suitable forms.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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