### ACCURATE DISCHARGE COEFFICIENT RELATIONSHIP FOR THE CRUMP WEIR

#### 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 *C _{d}* 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

*f*

_{1}and

*f*

_{2}are explicitly defined as simple and handy relationships. Compared to recent observations, the previous

*C*relationship causes a maximum deviation of only 0.864%, resulting in the same maximum deviation in the flow rate

_{d}*Q*computation. Therefore, it can be considered the most accurate and comprehensive

*C*relationship ever developed before for the Crump weir working under free overflow conditions. This allows the user to estimate the rate

_{d}*Q*sought with great certainty and confidence.

#### Keywords

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