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NOTCHES AND

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Originally appearing in Volume V14, Page 49 of the 1911 Encyclopedia Britannica.
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NOTCHES AND WEIRS § 41. Notches, Weirs and Byewashes.—A notch is an orifice ex-tending up to the free surface level in the reservoir from which the discharge takes place. A weir is a structure over which the water flows, the discharge being in the same conditions as for a notch. The formula of discharge for an orifice of this kind is ordinarily deduced by putting H1=o in the formula for the corresponding orifice, obtained as in the preceding section. Thus for a rectangular notch, put H1=o in (8). Then Q =, c BA/ (2g)H1, (II) where H is put for the depth to the crest of the weir or the bottom of the notch. Fig. 45 shows the mode in which the discharge occurs in the case of a rectangular notch or weir with a level crest. As, the free surface level falls very sensibly near the notch, the head H should be measured at some distance back from the notch, at a point where the velocity of the water is very small. Since the area of the notch opening is BH, the above formula is of the form Q =cXBH XkJ (2gH), where k is a factor depending on the form of the notch and expressing the ratio of the mean velocity of discharge to the velocity due to the depth H. § 42. Francis's Formula for Rectangular Notches.—The jet discharged through a rectangular notch has a section smaller than BH. (a) because of the fall of the water surface from the point where H h=ho+po/G—PIG ; Q =1 h2bs/ (2gh)dh = 3bJ2g lh'l:: —hie}, (6) where the first factor on the right is a coefficient depending on the form of the orifice. Now an orifice producing a rectangular jet must itself be very approximately rectangular. Let B be the breadth, H1, H2, the depths to the upper and lower edges of the orifice. Put b(h23—h4')/B(H22—HI=)=c. (7) Then the discharge, in terms of the dimensions of the orifice, instead of those of the jet, is Q = 3cBJ2g(Hz' -1-II'), (8) the formula commonly given for the discharge of rectangular orifices. The coefficient c is not, however, simply the coefficient of contraction, the value of which is b(h2—hi) /B(H2—HI), and not that given in (7). It cannot be assumed, therefore, that c in equation (8) is constant, and in fact it is found to vary for different values of B1H2 and B/H1, and must be ascertained experimentally. Relation between the Expressions (5) and (8).—For a rectangular is measured towards the weir, (b) in consequence of the crest con- I or, introducing a coefficient to allow for contraction, traction, (c) in consequence of the end contractions. It may be I Q= a_ t „cd~ (2g)H3, When a notch is used to gauge a stream of varying flow, the ratio B/H varies if the notch is rectangular, but is constant if the notch is triangular. This led Professor James Thomson to suspect that the coefficient of dis- charge, c, would be much more constant with different values of H in a triangular than in a rectangular notch, and this has been experimentally shown to be the case. FIG. 46. Hence a trian- gular notch is more suitable for accurate gaugings than a rectangular notch. For a sharp-edged triangular notch Professor J. Thomson found c=0.617. It will be seen, as in § 4t, that since 1BH is the area of section of the stream through the notch, the formula is again of the form Q=cXIBHXki/ (2gH), pointed out that while the diminution of the section of the jet due to the surface fall and to the crest contraction is proportional to the length of the weir, the end contractions have nearly the same effect whether the weir is wide or narrow. J. B. Francis's experiments showed that a perfect end contraction, when the heads varied from 3 to 24 in., and the length of the weir was not less than three times the head, diminished the effective length of the weir by an amount approximately equal to one-tenth of the head. Hence, if 1 is the length of the notch or weir, and H the head measured behind the weir where the water is nearly still, then the width of the jet passing through the notch would be l -o•2H, allowing for two end contractions. In a weir divided by posts there may be more than two end contractions. Q=3c(l-o•InH)H~I2gH =5.3Jc(l-o•InH)Hw. This is Francis's formula, in which the coefficient of discharge c is much more nearly constant for different values of 1 and It than in the ordinary formula. Francis found for c the mean value o•622, the weir being sharp-edged. § 43. Triangular Notch (fig. 46).-Consider a lamina issuing between the depths h and h+dh. Its area, neglecting contraction, will be bdh, and the velocity at that depth is sl (2gh). Hence the discharge for this lamina is b,l 2gh dh. But B/b=H/(H-h); b=B(H-h)/H. Hence discharge of lamina =B(H-h)J (2gh)dh/H; and total discharge of notch ('H =Q=Bs/ (2g)J o (H-h)hldh/H =s1B,/ (2g)H?.where k =1ffrT is the ratio of the mean velocity in the notch to the velocity at the depth H. It may easily be shown that for all notches the discharge can be expressed in this form. § 44. Weir with a Broad Sloping Crest.-Suppose a weir formed with a broad crest so sloped that the streams flowing over it have a movement sensibly rectilinear and uniform (fig. 47). Let the inner edge be so rounded as to prevent a crest contraction. Consider a filament aa', the point a being so far back from the weir that the velocity of approach is negligible. Let 00 he the surface level in the reservoir, and let a be at a height h" below 00, and h' above a'. Let h be the distance from 00 to the weir crest and e the thickness of the stream upon it. Neglecting atmospheric pressure, which has no influence, the pressure at a is Gh"; at a' it is Gz. If v be the velocity at a', v'/2g =h'+h" -z =h -e; Q=be Vlzg(h-e). Theory does not furnish a value for e, but Q=o for e=o and for e=h. Q has therefore a maximum for a value of e between o and h, obtained by equating dQ/de to zero. This gives e = ;h, and, inserting this value, Q=0.385 bh1/zgh, as a maximum value of the discharge with the conditions assigned. Experiment shows that the actual discharge is very approximately equal to this maximum, and the formula is more legitimately applicable to the discharge over broad-crested weirs and to cases such as the discharge with free upper surface through large masonry Coefficients for the Discharge over Weirs, derived from the Experiments of T. E. Blackwell. When more than one experiment was made with the same head, and the results were pretty uniform, the resulting coefficients are marked with an (*). The effect of the converging wing-boards is very strongly marked. Heads in Sharp Edge. Planks 2 in. thick, square on Crest. Crests 3 ft. wide. inches to ft. long, measured from still ft_ wing-boards 3 ft. long, 3 ft. long, 3 ft. long, 6 ft. long, 10 ft. long, ro ft. long, water in 3 long. 10 ft. long. 3 ft. long. 6 It, long. to ft. long. making an angle level. fall r in ts. fall 1 in 12. level. level. fall tin uS. g. Reservoir. of 6o°. t •677 .809 .467 '459 .435t '754 '452 545 '467 .381 '467 2 .675 .803 .509* .561 .585* .675 '482 '546 '533 '479* '495* 3 '630 .642* .563* .597* .569* '441 537 '539 '492* . 4 '617 .656 .549 .575 .602* .656 '419 '431 '455 '497* .• '515 5 •6o2 .650* •588 .hot* .609* -671 479 .516 .. .. .518 6 '593 •. 593* •608* .576* .. .501* . . •531 '507 '513 '543 7 .. .. •617* .608* •576* 488 513 527 497 8 .581 606* .590* .548* 470 491 .. .. .468 •507 9 530 •600 .569* .558* 476 492* '498 •480* •486 Io •614* '539 '534* .. '465* '455 12 '525 '534* 467* 1 4 .. .. '549* .. .. .. .. L 'The discharge per second varied from .461 to .665 cub. ft. in two experiments. The coefficient •435 is derived from the mean value. sluice openings than the ordinary weir formula for sharp-edged weirs. It should be remembered, however, that the friction on the sides and crest of the weir has been neglected, and that this tends to reduce a little the discharge. The formula is equivalent to the ordinary weir formula with c=0.577.
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