ACCURATE DISCHARGE COEFFICIENT RELATIONSHIP FOR THE CRUMP WEIR

B. ACHOUR, L. AMARA

Abstract


Due to its horizontal crest in the transverse direction and its inclined upstream and downstream faces, whose inclination is well defined, the Crump weir can be classified into the category of triangular longitudinal profile weirs, including the Bazin weir. It is an intermediate category between that which includes thin-crested weirs characterized by reduced thickness and that which includes broad-crested weirs that extend over a given length in the streamwise direction.

In practice, the Crump weir is preferably used as a sill for several reasons. The weir reduces the upstream flow velocity by raising the water level, which reduces or even avoids erosion. Additionally, the measurement of the upstream flow depth h, counted above the weir, is carried out with the greatest precision when the device is used as a flow meter.

The Crump weir as a flow measuring structure has not been studied from a theoretical point of view, and only experimental observations have enabled it to be calibrated. The resulting stage-discharge relationship is not only empirical but also incomplete since the effect of influential parameters, such as h/B, where B is the width of the rectangular approach channel, has not been accounted for, which affects the accuracy of the flow rate calculation. Only the effect of the relative elevation of the crest weir P* = P/h on the flow rate was examined on the basis of observations, where P is the elevation of the crest weir. The dimensionless parameter P* reflects the influence of the vertical contraction of the flow caused by the weir.

In this study, it is proven that the ratio h/B accounts for 23.5% as an average effect in the calculation of the discharge coefficient Cd and hence of the flow rate Q. The refined model describing this effect is yielded based on the analysis of observations available in the literature because current theories are unable to produce a mathematical representation of this effect. Unlike h/B, the effect of P* is derived from a rigorous theory based on the energy equation, judiciously transformed into dimensionless terms, along with rational hydraulic concepts. Therefore, the discharge coefficient relationship resulting from this study is a semiempirical formula that can be written symbolically in the following form . It is inferred that a is a constant whose appropriate value is estimated to be 0.8601, while the symbolic functions f1 and f2 are explicitly defined as simple and handy relationships. Compared to recent observations, the previous Cd relationship causes a maximum deviation of only 0.864%, resulting in the same maximum deviation in the flow rate Q computation. Therefore, it can be considered the most accurate and comprehensive Cd relationship ever developed before for the Crump weir working under free overflow conditions. This allows the user to estimate the rate Q sought with great certainty and confidence.


Keywords


Crump weir, stage-discharge relationship, flow measurement, semiempirical approach, discharge coefficient.

Full Text:

PDF

References


ABRAMOWITZ M., I.A. STEGUN I.A. (1972). Handbook of Mathematical Functions with Formula, graphs, and Mathematical Tables, 9th printing, 1972, p.14, New York: Dover.

ACHOUR B., AMARA L. (2022a). Compactness of Hydraulic Jump Rectangular Stilling Basins Using a Broad-Crested Sill, Larhyss Journal, No 51, pp. 31-41.

ACHOUR B., AMARA L. (2022b). Theoretical and experimental investigation of a lateral broad-crested contraction as a flow measurement device, Flow Measurement and Instrumentation, Vol. 86, Paper 102175.

ACHOUR B., AMARA L. (2022c). Flow measurement using a triangular broad crested weir theory and experimental validation, Flow Measurement and Instrumentation, Vol. 83, Paper 102088.

ACHOUR B., AMARA L. (2022d). Rectangular broad-crested flow meter with lateral contraction - Theory and experiment, Larhyss Journal, No 49, pp. 85-122.

ACHOUR B., BOUZIANE M.T., NEBBAR K. (2003). Broad-crested triangular flowmeter in rectangular channel, Larhyss Journal, No 2, pp. 7-43 (In French).

AFBLB, Agence Financière du Bassin Loire et Bretagne (1970). Book of special requirements for the production and approval of devices for measuring the flow of effluents, (In French).

AGRESTI A. (1990). Categorical Data Analysis, John Wiley and Sons, New York, USA.

AZIMI A.H., N. RAJARATNAM N. (2009). Discharge characteristics of weirs of finite crest length, journal of hydraulic engineering, vol. 135, pp. 1081-1085.

BIJANKHAN M., DI STEFANO C.D., FERRO V., KOUCHAKZDEH S. 2014). New Stage-Discharge Relationship for Weirs of Finite Crest Length, Technical Notes, Journal of Irrigation and Drainage Engineering, Vol. 140, Issue 3.

BOLSHAKOV V.A. (1984). Handbook of hydraulics, Kyiv, Vishcha school, 343p. (In Russian)

BOS M.G. (1976). Discharge Measurement Structures, Laboratorium Voor Hydraulica Aan Afvoerhydrologie, Landbouwhogeschool, Wageningen, The Netherlands.

BOS M.G. (1989). Discharge Measurement Structures, third ed., Publication 20, Int. Institute for Land Reclamation and Improvement, Wageningen, The Netherlands.

CHOW V.T. (1959). Open-Channel Hydraulics, McGraw Hill, New York, USA.

FILIPPOV Y.G., BRAKENI A. (2007). Use of weirs with sill of triangular profile for measurement of discharges of water in open streams and conduits,” in: Problems of Stable Development of Reclamation and Rational Resource Management: Proceedings International Science-Practice, Jubilee Conference (Kostyakov Session), Izd. VNIIA, Moscow, No 2, pp. 338-343 (In Russian).

HENDERSON F.M. (1966). Open Channel Flow, the McMillan Company, New York, N.Y, USA.

KHAFAGI A. (1942). Der Venturikanal: Theorie und Anwendung. Versuchsanstalt für Wasserbau, Eidgenössische Technische Hochschule Zürich, Mitteilung1. Leemann, Zürich, Switzerland, The venturi channel: theory and application. Laboratory for Hydraulic Engineering, Swiss Federal Institute of Technology in Zurich, Communication1, (in German).

LANGHAAR H.L. (1951). Dimensional Analysis and Theory of Models, John Wiley and Son Ltd, 1st Edition, 166p.

RAO N.S.G., MURALIDHAR D. (1963). Discharge characteristics of weirs of limit crest width, La Houille Blanche, No 18, pp. 537-545.

RF STATE STANDARD MI 2406–97. (1997). Discharge of Liquid in Gravity-flow Conduits of Water-Supply and Sewer Systems, A Technique of Carrying out Measurements by Means of Standard Weirs and Flumes, Izd. GNTs NIIVODGEO and AO IRVIS, Moscow, (In Russian).

SIA (1936). Contribution to the study of gauging methods, Bulletin 18, Schw. Wasserforschung, Bern, Switzerland, (In French).

SPIEGEL M.R. (1974). Mathematical Handbook of Formulas and Tables, 20th Edition, McGraw Hill Inc, New York, USA.

VOGT W.P., Johnson R.B. (2015). Dictionary of Statistics & methodology: A Nontechnical Guide for the Social Sciences, 5th Edition, SAGE Publishing.

ZUIKOV A.L. (2017). Hydraulics of the classical Crump weir water gauge, Power Technology and Engineering, Vol. 50, Issue 6, pp. 50-59.


Refbacks

  • There are currently no refbacks.


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.