Discharge characteristics of triangular-notch thin-plate weirs

The triangular-notch, thin-plate weir is a convenient, inexpensive, and relatively precise flow-measuring instrument. It is frequently used to measure the flow of water in laboratories and in small, natural streams. This report includes an extensive review of the literature and presents a comprehensive analysis of the discharge characteristics of triangular-notch weirs. Previously published data are analyzed in the light of the effective-head concept, and a new discharge formula is proposed. Coefficients are recommended and requirements for precise measurements are described. Limits of applicability are discussed. INTRODUCTION The triangular-notch,thin-plate weir is used widely for measuring the flow of liquids in flumes and open channels. Simple in design and easily made from readily available materials, it is inexpensive, convenient to use, and easy to maintain. In permanent or portable form it is frequently used to measure the flow of water in laboratories and in small, natural streams. When several forms of weirs or flumes might be used, the triangular-notch weir is often preferred because of its greater accuracy at low flows or its lesser sensitivity to approach-channel geometry and velocity distribution. Within the range of conditions for which verification data are adequate, and with reasonable care in its construction, installation, and use, the triangular-notch, thin-plate weir is a relatively precise instrument. The triangular-notch weir has been the subject of considerable experimental research and published discussion. Unfortunately, however, most of the laboratory investigations have been restricted to a narrow range of notch angles and channel geometries. The 90°-notch weir has been most extensively studied. With few exceptions, water has been the liquid used in laboratory tests. A large number of empirical discharge formulas have been proposed for triangular-notch weirs. Most of these were designed to fit a parBl B2 FLOW OF WATER OVER WEIRS AND DAMS ticular set of experimental data. None provides a comprehensive solution, even for a single liquid. Deficiencies in these formulas are concealed with numerous limits of applicability which greatly restrict their usefulness. This report presents a comprehensive analysis of the discharge characteristics of triangular-notch, thin-plated weirs. Previously published data from various sources have been reanalyzed, and some of the traditional discharge formulas are compared with an original solution. The report is based on one of a series of studies of weirs and spillways which was undertaken by the U.S. Geological Survey under the direction of C. E. Kindsvater. Previously published reports in the series are concerned with broad-crested weirs (Tracy, 1957), rectangular, thin-plate weirs (Kindsvater and Carter, 1959), and embankment-shaped weirs (Kindsvater, 1964). Portions of this report are adapted from unpublished reports and drafts of weir standards prepared by C. E. Kindsvater. DESCRIPTION OF THE WEIR The weir which is the subject of this study is a symmetrical, V-shaped notch in a vertical thin plate. The line which bisects the angle of the notch is vertical and equidistant from the sides of the approach channel. The weir plate is smooth, plane, and perpendicular to the sides as well as the bottom of the approach channel. Figure 1 shows a triangular-notch, thin-plate weir installed at the end of a rectangular channel. The crest surfaces of the weir notch are plane surfaces which form sharp, right-angle corners at their intersection with the upstream face of the weir plate. The width of the crest surface varies, but it is generally between 1/32 and 1/16 inch. If the weir plate is thicker than 1/16 inch, the downstream edges of the notch are chamfered to make an angle of not less than 45° with the surface of the crest. Ideally, the channel upstream from the weir is straight, smooth, horizontal, rectangular, and of sufficient length to develop the normal (uniform flow) turbulence and velocity distribution for all discharges. Usually, however, it is less than ideal, and baffles or screens are provided in order to simulate a normal velocity distribution. Channel and tailwater conditions downstream from the weir are such as to permit a free, fully ventilated flow from the notch. Provisions for ventilation ensure that pressure on the nappe surfaces is atmospheric. The tailwater is low enough that it does not interfere with the ventilation of the nappe or free flow from the notch. The head on the weir is the measured vertical distance from the water surface to the vertex of the notch. The head-measuring section is located a sufficient distance upstream from the weir to avoid the region of surface draw-down, and it is sufficiently close to the weir that the DISCHARGE OVER TRIANGULAR-NOTCH THIN-PLATE WEIRS B3 Upstream face of weir plate ^45° FIGURE l.-The triangular-notch, thin-plate weir. energy loss between the measuring section and the weir is negligible. A distance of 4 to 5 h is recommended. BASIC EQUATION OF DISCHARGE The traditional equation of discharge for triangular-notch weirs is derived on the basis of an assumed analogy between the weir and the B4 FLOW OF WATER OVER WEIRS AND DAMS orifice. In the derivation, an approximate velocity equation is integrated over assumed area limits of the nappe in the plane of the weir. The result is an equation which is useful mainly because it is dimensionally correct and because it is used almost universally as the basis for the analysis of experimental data. It is used herein as the basic equation of discharge. In its traditional form, the basic discharge equation is Q=CV20 tan *60 (1) lo £ in which Q is the volume rate of flow or discharge in cubic feet per second; C is the nondimensional coefficient of discharge; g is the acceleration due to gravity in feet per second per second; 0 is the angle included between the sides of the notch, usually measured in degrees; and h is the potentiometric head or height of the upstream liquid surface measured with respect to the vertex of the notch, in feet (in metric units, discharge would be stated in cubic meters per second, head in meters, and g in meters per second per second; C, being nondimensional, would remain unchanged in numerical value). Whereas C is assumed to be a coefficient of contraction in the traditional derivation, in actuality it is an experimentally determined coefficient which is dependent upon all the variables needed to describe the channel, the weir, and the discharging liquid. Thus, in the absence of a rigorous theoretical solution, dimensional relations must be used to guide the analysis of experimental data on triangular-notch weirs. DIMENSIONAL ANALYSIS The principal variables needed to define the discharge characteristics of a triangular-notch, thin-plate weir in a rectangular channel (fig. 1) are: Q, the discharge; B, the width of the approach channel; P, the height of the notch vertex with respect to the floor of the approach channel; h, the head on the weir, referred to the vertex of the notch; 6, the angle included between the sides of the notch; p, the density of the liquid; n, the viscosity of the liquid; a, the surface tension of the liquid; and 7, the specific weight of the liquid. If the discharge is selected as the dependent variable, a complete statement of the discharge function is Q-f9t p 9 pt o,y). (2) From this equation a nondimensional discharge ratio can be expressed as a function of five nondimensional ratios,