A generalized mathematical theory and experimental verification of proportional notches

Keshava Murthy, K. ; Seshagiri, N. (1968) A generalized mathematical theory and experimental verification of proportional notches Journal of the Franklin Institute, 285 (5). pp. 347-363. ISSN 0016-0032

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Related URL: http://dx.doi.org/10.1016/0016-0032(68)90483-3


A theory and generalized synthesis procedure is advocated for the design of weir notches and orifice-notches having a base in any given shape, to a depth a, such that the discharge through it is proportional to any singular monotonically-increasing function of the depth of flow measured above a certain datum. The problem is reduced to finding an exact solution of a Volterra integral equation in Abel form. The maximization of the depth of the datum below the crest of the notch is investigated. Proof is given that for a weir notch made out of one continuous curve, and for a flow proportional to the mth power of the head, it is impossible to bring the datum lower than (2m - 1)a below the crest of the notch. A new concept of an orifice-notch, having discontinuity in the curve and a division of flow into two distinct portions, is presented. The division of flow is shown to have a beneficial effect in reducing the datum below (2m - 1)a from the crest of the weir and still maintaining the proportionality of the flow. Experimental proof with one such orifice-notch is found to have a constant coefficient of discharge of 0.625. The importance of this analysis in the design of grit chambers is emphasized.

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