WEIRS A weir is a dam or obstruction placed across a stream for the purpose of diverting the water and causing it to flow through a channel of known dimensions, which channel may be a notch or opening in the obstruction itself. The notch is usually rectangular in form. There are two general types of weirs, namely, those with end contractions, as in Fig. 1 (a), and those without end contractions, as in Fig. 1 (b). Crest of the Weir.-The edge of the notch over which the water flows, as shown in cross-section at a, Fig. 1 (c) and (d), is (c) (d) (b) called the crest of the weir. FIG. 1 In all weirs, the inner edge of the crest is made sharp, so that, in passing over it, the water touches along a line. The same statement applies to the inner edge of both the top and the ends of the notch in weirs with end contractions. For very accurate work, the edges of the notch should be made with a thin plate of metal having a sharp inner edge, as shown in Fig. 1 (d); but for ordinary work the edges of the board in which the notch is cut may be chamfered off to an angle of about 30°, as shown at (c). The top edge of the notch must be straight and set perfectly level, and the sides must be set carefully at right angles to the top. Means for admitting air under the falling sheet of water must be made; otherwise, there will be formed a partial vacuum that tends to increase the discharge. The sides of a weir without end contractions should be smooth and straight and should project a slight distance beyond the crest. Standard Dimensions for Weirs.-The distance from the crest of the weir to the bottom of the feeding canal or reservoir should be at least three times the head H, Fig. 1 (c) and (d); and, with a weir having end contractions, the distance from the vertical edges to the sides of the canal should also be at least three times the head. The water must approach the weir quietly and with little velocity; theoretically, it should have no velocity. It is often necessary to place one or more sets of baffle boards or planks across the stream at right angles to the flow, and at varying depths from the surface, to reduce the velocity of the water as it approaches the weir. Theoretical and Actual Discharge of Weirs.-The theoretical discharge of a weir is Q=5.347 bH} in which b is the length of the crest and H is the effective depth producing the discharge. When the velocity of approach is inappreciable, the effective depth is the distance from the crest of the weir to the surface of the water at a point up-stream beyond the curve assumed by the flowing water as it approaches the weir; but, when the velocity of approach v is considerable, Q=5.347 b (H+h)} in which h= v2 The actual discharge of weirs is, when the velocity of approach is not considered, In these formulas n denotes the number of end contractions; hence, for a weir with two end contractions, n=2; for a weir with one end contraction, n = 1; and for a weir with no end conIn the last case, the two preceding formulas tractions, n=0. become, respectively, and Q=3.33 bH Q=3.336 [(H+h)*— h}] The velocity of approach can be determined by first finding Q from the formula Q=3.33 bH, and dividing it by the area of the cross-section of the channel; the quotient will be the velocity of approach v and h will equal .01555 v2. EXAMPLE 1.-A weir with end contractions is 5 ft. long and the measured head is .872 ft. Calculate the discharge on the assumption that the velocity of approach is negligible. SOLUTION. Substituting the given values in the proper formula, Q=3.33X(5-X.872) X.872-13.085 cu. ft. per sec. The preceding formulas are known as Francis's formulas and are recommended for heads from 5 to 19 in. For lower heads, the formula of Fteley and Stearns, which follows, is recommended: Q=3.316(H+1.5h)++.007b For higher heads, Bazin's formula is recommended: The last two formulas are applicable only to weirs with no end contractions. In these formulas, p is the distance from the bottom of the channel to the crest; the other letters have the same significance as before. discharge of a triangular weir whose effective head is 9 in. SOLUTION. Substituting the given values in the formula, Q=2.54X.75=1.24 cu. ft. per sec. FLOW OF WATER IN CHANNELS A channel is the bed of a long body of water flowing under the action of gravity. An artificial channel whose bed is formed by the natural soil is called a canal, and when the bed is artificial, like a flume or a sewer pipe, it is called a conduit. A ditch is a small canal. The slope s of a channel is the ratio of the fall h to the length I in which the fall occurs; or h S= เ The wetted perimeter of a cross-section of a channel is the part of the boundary in contact with the water. The hydraulic radius of a channel is the ratio of the area of the cross-section of the water in the channel to the wetted perimeter. Chezy's Formula.-The fundamental formula for the velocity of flow in a channel is v=c √rs, in which s is the slope of the channel; r, the hydraulic radius; and c, a variable coefficient whose value is given by Kutter's formula, which is, 1 .00155 23+ + In this formula n is the coefficient of roughness, whose values are as follows: Character of Channel Clean well-planed timber.. Value of n .009 .011 Clean, smooth, glazed iron and stoneware pipes. . .010 .012 .025 Coarse rubble masonry and firm compact gravel. .020 with vegetation and strewn with stones and other detritus, according to condition. .035 to .050 EXAMPLE. Find the discharge of a rough-plank sluice 24 in. wide, when the depth of the water in the sluice is 15 in. and the fall 3 in. in 100 ft. SOLUTION. The slope s=.25÷100= .0025; the wetted perimeter p=2+(2×1.25) = 4.5 ft.; and the area A of the water cross-section=2X1.25=2.5 sq. ft. The hydraulic radius is, The value of n for unplaned therefore, r=2.5÷4.5.5556. timber is .012; therefore, Substituting the values found in Chezy's formula, v = 114.7 V.5556 X.0025=4.27 ft. per sec. Therefore, the discharge is Q=Av=2.5X4.27 = 10.675 cu. ft. per sec. Discharge of Large Streams.-The discharge of a large body of water, when it is impracticable to construct a weir, is determined by measuring, on one hand, the mean velocity v at a cross-section of flowing water by means of floats or by the use of special instruments, and, on the other hand, by ascertaining the area A of that cross-section. Then, the discharge Q = Av The current meter affords the most convenient and accurate method of measuring velocities of a stream. One form of this instrument is shown in Fig. 1. The number of revolutions of the buckets b depending on the velocity of the flow is recorded electrically on the dials m and n. The relation between this |