HYDRAULICS.] HYDROMECHANICS 473 Coefficients for the Discharge over Weirs, derived from the Experiments of Mr Blackucdl. What more than one experiment was made with tlie same head, and the results were pretty uniform, the resulting coefficients are marked with an (*). The effect of the con- verging wing-boards is very strongly marked. Heads in inches Sharp Edge. Planks 2 inches thick, square on Crest Crests 3 feet wide. measured from still Water in Reservoir. 3 feet lung. 10 feet long. 3 feet long. G feet long. 10 feet long. wing boards making an angle of 60. 3 feet long, level. 3 feet long, fail 1 in 18. 3 feet long, fall 1 in 12. 6 feet long, level. 10 feet long, level. 10 feet lung, fall 1 in 18. 1 (377 809 467 459 435 1 754 452 545 467 381 467 2 675 803 509* 561 585* 675 482 546 533 479* 495* 3 630 642* 563* 597* 569* 441 537 539 492* 4 G17 656 549 575 602* 656 419 431 455 497* 515 5 G02 650* 588 601* 609* 671 479 516 518 G 593 593* 608* -576* 501* 531 507 513 543 t 617* 608* 576* 488 513 527 497 u 581 606* 590* 548* 470 491 468 -507
530 600 569* 558* 476 492* 498 480* 486 10 614* 539 534* 465* 455 12 525 534* 467* li 549* 1 The discharge per second varied from 401 to "665 cubic feet in two experiments. The coefficient 43-5 is derived from the mean value. SPECIAL CASES OF DISCHARGE FROM ORIFICES. 41. Cases in which the, Velocity of Approach needs to be taken into Account. Rectangular Orifices and Notches. In finding the velocity at the orifice in the preceding investigations, it has been assumed that the head h lias been measured from the free surface of still water above the orifice. In many cases which occur in practice the channel of approach to an orifice or notch is not so large, relatively to the stream through the orifice or notch, that the velocity in it can be disregarded. Let h lt 7i 2 (fig. 51) be the heads measured from the free surface to the top and bottom edges of a rectangular orifice, at a point in the Fig. 51. channel of approach where the velocity is u. fall of the free surface, It is obvious that a has been somewhere expended in producing the velocity u, and hence the true heads measured in still water would have beenAj + f) and 7i 2 + fj. Consequently the discharge, allowing for the velocity of approach, is +D -fo+j) .... (i). And for a rectangular notch for which Ti = 0, the discharge is f (2). In cases where u can be directly determined, these formuhe give the discharge quite simply. When, however, u is only known as a function of the section of the stream in the channel of approach, they become complicated. Let 1 be the sectional area of the channel where h l and h., are measured. Then u = Q- and f) = -^l This value introduced in the equations above would render them excessively cumbrous. In cases therefore where fl only is known, it is best to proceed by approximation. Calculate an approximate value Q of Q by the equation Then )=~^ nearly. This value of f) introduced in the equations above will give a second and much more approximate value of Q. 42. Partially Submerged Rectangular Orifices and Notches. When the tail water is above the lower but below the upper edge of the orifice, the flow in the two parts of the orifice, into which it is divided by the surface of the tail water, takes place under differ ent conditions. A filament M^ (fig. 52) in the upper part of the orifice issues with a head h which may have any value between /tj and/i. But a filament M..,m. 2 issuing in the lower part of the orifice has a velocity due to h" - h" , or h, simply. In the upper part Fig. 52. of the orifice the head is variable, in the lower constant. If Q 1; Q 2 are the discharges from the upper and lower parts of the orifice", b the width of the orifice, then In the case of a rectangular notch or weir, 7^ 1 = 0. Inserting this value, and adding the two portions of the discharge together, ve get for a drowned weir where h is the difference of level of the head and tail water, and h* is the head from the free surface above the weir to the weir crest. If velocity of approach is taken into account, let Ij be the head due to that velocity ; then, adding I) to each of the heads in the equa tions (3), and reducing, we get for a weir Q = cb V2^ [(h, + )) ( h + )T - 4 (h an equation which may be useful in estimating flood discharges. Bridge Piers and other Obstructions in Streams. When the piers of a bridge are erected in a stream they create an obstruction to the flow of the stream, Avhich causes a difference of surface-level above and below the pier (fig. 53). If it is necessary to estimate this differ ence of level, the ilow between the piers may be treated as if it occurred over a drowned weir. But the value of c in this case is imperfectly known.
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