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SPECIAL CASES OF DISCHARGE FROM
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See also:SPECIAL CASES OF See also:DISCHARGE FROM ORIFICES § 45. Cases in which the Velocity of Approach needs to be taken into See also:Account. Rectangular Orifices and Notches.—In finding the velocity at the orifice in the preceding investigations, it has been assumed that the See also:head h has been measured from the See also:free See also:surface of still See also: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 hi, h2 (fig. 48) be the heads measured from the free surface to the See also:top and bottom edges of a rectangular orifice, at a point in the channel of approach where the velocity is u. It is obvious that a fall of the free surface, = u2/2g has been somewhere expended in producing the velocity u, and hence the true heads measured in still water would have been hi+f and h2+h. Consequently the discharge, allowing for the velocity of approach, is Q=acbd2gl(h2+I))—(hl+TO • (I) And for a rectangular notch for which hi =o, the discharge is Q= acbd 2gl(hs + 9)i-hi]. (2) In cases where u can be directly determined, these formulae give the discharge quite simply. When, however, u is only known as a See also:function of the See also:section of the stream in the channel of approach, they become complicated. Let 12 be the sectional See also:area of the channel where h, and 112 are measured. Then u=Q/ 12 and 1)=Q2/2g See also:S22. This value introduced in the equations above would render them excessively cumbrous. In cases therefore where S2 only is known, it is best to proceed by approximation. Calculate an approximate value Q' of Q by the See also:equation Q'= 3cbd 2g]hhi—h0]. Then I) = Q'2/2gl-22 nearly. This value of 1) introduced in the equations above will give a second and much more approximate value of Q. § 46. Partially Submerged Rectangular Orifices and Notches.—When the tail water is above the See also: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 See also:place under different conditions. A filament M,mi (fig. 49) in the upper See also:part of the orifice issues with a head h' which may have any value between h, and h. But a filament M2m2 issuing in the lower part of the orifice has a velocity due to h"—h"', or h, simply. In the upper part of the orifice the head is variable, in the lower See also:constant. If Qi, Q2 are the discharges from the upper and lower parts of the orifice, b the width of the orifice, then Q1=Jcb-N/2glhi—See also:hit] Q2=cb(h2—h)d2gh In the See also:case of a rectangular notch or See also:weir, hi =0. Inserting this value, and adding the two portions of the discharge together, we get for a drowned weir Q =cbd2gh(h2—h/3), (4) where h is the difference of level of the head and tail water, and h2 is the head from the free surface above the weir to the weir See also:crest (fig. 5o). From some experiments by Messrs A. Fteley and F. P. Stearns (Trans. Am. See also:Soc. C.E., 1883, p. 102) some values of the coefficient c can be reduced h3/h2 c' h3/h2 c o•I 0.629 0.7 0.578 0.2 0.614 o•8 0.583 0.3 o•600 0.9 0.596 0.4 0.590 0.95 0.607 0.5 0.582 I•oo o•628 o•6 0.578 If velocity of approach is taken into account, let ') be the head due to that velocity; then, adding f) to each of the heads in the equations (3), and reducing, we get for a weir Q=cbd2g[(h2+13) (h+13)4—a(h+1))i_*17i]; (5) an equation which may be useful in estimating See also:flood discharges. See also:Bridge Piers and other Obstructions in Streams.—When the piers of a bridge are erected in a stream they create an obstruction to the value of c in this case is oiiai~i i~~%_~~~i~~~Tiiis~/i~~i~•~~ - imperfectly known. FIG. 50. § 47. See also:Bazin's Researches on Weirs.—H. Bazin has executed a See also:long See also:series of researches on the flow over weirs, so systematic and See also:complete that they almost supersede other observations. The account of them is contained in a series of papers in the Annales See also:des Ponts et Chaussees (See also:October 1888, See also:January 189o, See also:November 1891, See also:February 1894, See also:December 1896, 2nd trimestre 1898). Only a very abbreviated account can be given here. The See also:general See also:plan of the experiments was to establish first the coefficients of discharge for a See also:standard weir without end contractions; next to establish weirs of other types in series with the standard weir on a channel with steady flow, to compare the observed heads on the different weirs and to determine their coefficients from the discharge computed at the standard weir. A channel was constructed parallel to the See also:Canal de Bourgogne, taking water from it through three sluices 0.3 X I •o metres. The water enters a See also:masonry chamber 15 metres long by 4 metres wide where it is stilled and passes into the canal at the end of which is the standard weir. The canal has a length of 15 metres, a width of 2 metres and a See also:depth of i•6 metres. From this extends a channel 200 metres in length with a slope of I mm. per See also:metre. The channel is 2 metres wide with See also:vertical sides. The channels were constructed of See also:concrete rendered with See also:cement. The water levels were taken in See also:chambers constructed near the canal, by floats actuating an See also:index on a See also:dial. See also:Hook gauges were used in determining the heads on the weirs. Standard Wcir.—The weir crest was 3.72 ft. above the bottom of the canal and formed by a See also:plate 4 in. thick. It was See also:sharp-edged with free overfall. It was as wide as the canal so that end See also:con-tractions were suppressed, and enlargements were formed below the crest to admit See also:air under the water See also:sheet. The channel below the weir was used as a gauging tank. Gaugings were made with the weir 2 metres in length and afterwards with the weir reduced to 1 metre and 0.5 metre in length, the end contractions being sup-pressed in all cases. Assuming the general See also:formula Q=mlhd (2gh), (I) hZ . . Ma i- i I my (3) flow of the stream, which causes a difference of surface-level above and below the See also:pier (fig. 51). If it is neces- h, sary to estimate this difference of level, the flow between the piers may be treated as if it occurred over a drowned weir. But the Bazin arrives at the following values of m: Coefficients of Discharge of Standard Weir. Head h metres. Head h feet. m 0'05 .164 0.4485 o•I0 .328 0.4336 0.15 .492 0.4284 0.20 •656 0.4262 0.25 •82o 0.4259 0.30 •984 0.4266 0'35 1.148 0.4275 0.40 1.312 0.4286 0'45 1'476 0.4299 0.50 1.64o 0.4313 0'55 I.8o4 0.4327 o•6o 1.968 0.4341 Bazin compares his results with those of Fteley and Stearns in 1877 and 1879, correcting for a different velocity of approach, and finds a See also:close agreement. See also:Influence of Velocity of A pproach.-To take account of the velocity of approach u it is usual to replace It in the formula by h+au2/2g where a is a coefficient not very well ascertained. Then Q =µl (h+au2/2g) .J [2g(h+au2/2g)} =µlh-) (2gh) (1+ au2/2gh)3. The See also:original See also:simple equation can be used if m= µ(I +au2/2gh)l or very approximately, since u2/agh is small, m =µ (1 +' au2/2gh) • (3) Now if p is the height of the weir crest above the bottom of the canal (fig. 52), u =Q/l(p+h). Replacing Q by its value in (I) u2/2gh = Q2/ f 2gh12 (p +h)2} =m2{h/(p+h)}2, (4) so that (3) may be written m =µ[I -{-k{h/(p+h) }2l. (5) Gaugings were made with 7////////////, weirs of 0.75, 0.50, 0.35, and Fin. 52. 0.24 metres height above the canal bottom and the results compared with those of the standard weir taken at the same See also:time. The discussion of the results leads to the following values of m in the general equation (I) :- m =µ(1 +2.5u2/2gh) =µ[ 1 +0•55{h/(P+h)12l. Head h metres. Head h feet. µ 0.05 . .164 0.4481 0.10 .328 0.4322 0.20 .656 0.4215 0.30 •984 0'4174 0.40 } 1.312 0.4144 0.50 1.64o 0.4118 o•6o 1.968 0.4092 An approximate formula forµ is: µ=0.405+0.003/h (h in metres) µ=0•405+o•oI/h (h in feet). Inclined Weirs.-Experiments were made in which the See also:plank weir was inclined up or down stream, the crest being sharp and the end contraction suppressed. The following are coefficients by which the discharge of a vertical weir should be multiplied to obtain the discharge of the inclined weir. Inclination up stream Coefficient. t o t 0.93 3 to 2 0.94 3 to 1 o•96 I•oo 3 to I 1.04 3 to 2 I•o7 1 to 1 I.10 1 to 2 I.12 I to 4 1.09 The coefficient varies appreciably, if h/p approaches unity, which case should be avoided. In all the preceding cases the sheet passing over the weir is detached completely from the weir and its under-surface is subject to atmospheric pressure. These conditions permit the most exact determination of the coefficient of discharge. If the sides of the canal below the weir are not so arranged as to permit the See also:access of air under the sheet, the phenomena are more complicated. So long as the head does not exceed a certain limit the sheet is detached [DISCHARGE FROM ORIFICES from the weir, but encloses a See also:volume of air which is at less than atmospheric pressure, and the tail water rises under the sheet. The discharge is a little greater than for free overfall. At greater head the air disappears from below the sheet and the sheet is said to be " drowned." The drowned sheet may be See also:independent of the tail water level or influenced by it. In the former case the fall is followed by a rapid, terminating in a See also:standing See also:wave. In the latter case when the See also:foot of the sheet is drowned the level = _ = °-__ :. of the tail water influences the discharge even if it is below the weir crest. Weirs with See also:Flat Crests.- The water sheet may See also:spring o P or may clear from addhheere up to stream the e flat edge J/////////1////j/////~f////, . // crest falling free beyond the FIG. 53. downstream edge. In the former case the See also:condition is that of a sharp-edged weir and it is realized when the head is at least See also:double the width of crest. It may arise if the head is at least is the width of crest. Between these limits the condition of the sheet is unstable. When the sheet is adherent the coefficient m depends on the ratio of the head h to the width of crest c (fig. 53), and is given by the equation m=mi [0.70+o•185h/cl, where m1 is the coefficient for a sharp-edged weir in similar con- ditions. Rounding the up- stream edge even to a small h extent modifies the dis- See also:charge. If R is the See also:radius of the rounding the co- efficient m is increased in the ratio 1 to t +R/h nearly. The results are limited to R less than '` in. Drowned Weirs.-Let h (fig. 54) be the height of FIG. 54• head water and hi that of tail water above the weir crest. Then Bazin obtains as the appro]dmate formula for the coefficient of discharge m=I.05m,[I+ih1/p]-1{ (h-hl)/h}, where as before ml is the coefficient for a sharp-edged weir in similar conditions, that is, when the sheet is free and the weir of the same height. § 48. Separating Weirs. - Many towns derive their water-See also:supply from streams in high moorland districts, in which the flow is extremely variable. The water is collected in large storage reservoirs, from which an See also:uniform supply can be sent to the See also:town. In such cases it is desirable to See also:separate the coloured water which comes down the streams in high floods from the purer water of See also:ordinary flow. The latter is sent into the reservoirs; the former is allowed (2) Values of µ- Vertical weir . . . Inclination down stream See also:HYDRAULICS to flow away down the original stream channel, or is stored in separate reservoirs and used as See also:compensation water. To accomplish the separation of the flood and ordinary water, See also:advantage is taken of the different See also:horizontal range of the parabolic path of the water falling over a weir, as the depth on the weir and, consequently, the velocity See also:change. Fig. 55 shows one of these separating weirs in the See also:form in which they were first introduced on the See also:Manchester Water-See also:works; fig. 56 a more See also:modern weir of the same See also:kind designed by See also:Sir A. Binnie for the See also:Bradford Waterworks. When the quantity of water coming down the stream is not excessive, it drops over the weir into a transverse channel leading to the reservoirs. In flood, the water springs over the mouth of this channel and is led into a See also:waste channel. It may be assumed, probably with accuracy enough for See also:practical purposes, that the particles describe the parabolas due to the mean velocity of the water passing over the weir, that is, to a velocity Is/ (2gh), where h is the head above the crest of the weir. Let cb=x be the width of the orifice and ac=y the difference of level of its edges (fig. 57). Then, if a particle passes from a to b in t seconds, y=zgt2, x=iJ(2gh)t; • y = 6x2/h, which gives the width x for any given difference of level y and head h, which the See also:jet will just pass over the orifice. Set off ad vertically d 3 v h and equal to Zg on any See also:scale; af horizontally and equal to a J (gh). See also:Divide af, fe into an equal number of equal parts. Join a with the divisions on ef. The intersections of these lines with verticals from the divisions on af give the parabolic path of the jet. MOUTHPIECES—HEAD CONSTANT § 49. Cylindrical Mouthpieces.—When water issues from a See also:short cylindrical See also:pipe or See also:mouthpiece of a length at least equal to 11 times its smallest transverse See also:dimension, the stream, after contraction within the mouthpiece, expands to fill it and issues full See also:bore, or without contraction, at the point of discharge. The discharge is found to be about one-third greater than that from a simple orifice of the same See also:size. On the other See also:hand, the See also:energy of the fluid per unit of See also:weight is less than that of the stream from a simple orifice with the same head, because part of the energy is wasted in eddies produced at the point where the stream expands to fill the mouthpiece, the See also:action being something like that which occurs at an abrupt change of section. Let fig. 58 represent a See also:vessel discharging through a cylindrical mouthpiece at the depth h from the free surface, and let the See also:axis of the jet XX be taken as the datum with reference to which the head is estimated. Let it be the area of the mouthpiece, w the area of the stream at the contracted section EF. Let v, p be the velocity and pressure at EF, and ran pr the same quantities at GH. If the discharge is into the air, pr is equal to the atmospheric pressure P. The See also:total head of any filament which goes to form the jet, takenat a point where its velocity is sensibly zero, is h+p°/G; at EF the total head is v2/2g+p/G; at GH it is vr2/2g+pi/G. Between EF and GH there is a loss of head due to abrupt change of velocity, which from eq. (3), § 36, may have the value (v —vi)2/2g. Adding this head lost to the head at GH, before equating it to the heads at EF and at the point where the filaments start into See also:motion,— h +p°/G =v2/2g+p/G =v12/2g+p,/G+(v -vr)2/2g. But wv=l2vr, and w=c,12, if c° is the coefficient of contraction within the mouthpiece. Hence v=S2v,/w=v,/c,. Supposing the discharge into the air, so that pr=p°, h+p°/G=vr2/2g+p°/G±(v,2/2g) (See also:tic, -1)2; (v,/?g) iI+(I/c°—I)2}=h; v, = J (2gh)l J 11 + (I/cc — 1)2}; (I) where the coefficient on the right is evidently the coefficient of velocity for the cylindrical mouthpiece in terms of the coefficient of con-See also:traction at EF. Let c°=o.64, the value for simple orifices, then the coefficient of velocity is cv=l/J II+(I/e°—I)"} =o'87 (2) The actual value of c„ found by experiment is o•82, which does not differ more from the theoretical value than might be expected if the See also:friction of the FIG. 58. mouthpiece is allowed for. Hence, for mouthpieces of this kind, and for the section at GH, cv=0.82 c,=I•oo c=0.82, Q=o•82t2J (2gh). It is easy to see from the equations that the pressure p at EF is less than atmospheric ppressure. Eliminating v1, we get (p° p)/G=ih nearly; (3) or p=p°—IGhlb per sq. ft. If a pipe connected with a See also:reservoir on a lower level is introduced into the mouthpiece at the part where the contraction is formed (fig. 59), the water will rise in this pipe to a height KL = (p° —p) /G = 4h nearly. If the distance X is less than this, the water from the lower reservoir will be forced continuously into the jet by the atmospheric pressure, and discharged with it. This is the crudest form of a kind of See also:pump known as the jet pump. § 5o. Convergent Mouthpieces.—With convergent mouthpieces there is a contraction within the mouthpiece causing a loss of head, and a diminution of the velocity of discharge, as with cylindrical mouthpieces. There is also a second contraction of the stream out-See also:side the mouthpiece. Hence the discharge is given by an equation of the form Q —czc,SzJ (2gh), (4) where S2 is the area of the See also:external end of the mouthpiece, and ccfl the section of the contracted jet beyond the mouthpiece. Convergent Mouthpieces (See also:Castel's Experiments).—Smallest See also:diameter of orifice =o.o5085 ft. Length of mouthpiece 2.6 Diameters. See also:Angle of Coefficient of Coefficient of Coefficient of Convergence. Contraction, Velocity, Discharge, c, c„ c 0° 0' 999 '830 •829 1° 36' 1.000 •866 •866 3° 10' r.001 •894 '895 4° 10' I.002 .910 •912 5° 26' I.004 .920 •924 7° 52' '998 931 •929 8° 58' 992 '942 '934 ro° 20' •987 '950 •938 12° 4' .986 •955 •942 - 13° 24' .983 •962 '946 14° 28' '979 .966 •941 16° 36' .969 971 '938 19° 28' '953 970 '924 21° o' 945 971 •918 23° 0' 937 '974 '913 29° 58' '919 '975 .896 40° 20' .887 .98o .869 48° 50' •861 .984 '847 The maximum coefficient of discharge is that for a mouthpiece with a convergence of 13° 24'. A. 4/Z//////P/A( ` "Ai —~ A : G ~2 The values of co and co must here be determined by experiment. The above table gives values sufficient for practical purposes. Since the contraction beyond the mouthpiece increases with the convergence,or, what is the same thing, co diminishes, and on the other hand the loss of energy diminishes, so that co increases with the convergence, there is an angle for which the product c,, co, and consequently the discharge, .is a maximum. § 51. Divergent Conoidal Mouthpiece.—Suppose a mouthpiece so designed that there is no abrupt change in the section or velocity of the stream passing through it. It may have a form at the inner end approximately the same as that of a simple contracted vein, and may then enlarge gradually, as shown in fig. 6o. Suppose that at EF it becomes cylindrical, so that the jet may be taken to be of the diameter EF. Let w, v, p be the section, velocity and pressure at CD, and 12, v1, pi the same quantities at EF, pa being as usual the atmospheric pressure, or pressure on the free surface AB. Then, since there is no loss of energy, except the small frictional resistance of the surface of the mouthpiece, h+pa/G =v2/2g+p/G =See also:vie/2g+pi/G. If the jet discharges into the air, pi = pa ; and v,2/2g =h; vi = (2gh) ; or, if a coefficient is introduced to allow for friction, v~ = co J (2gh) ; where co is about 0.97 if the mouthpiece is smooth and well formed. Q = !hi = c„st,l (2gh). Hence the discharge depends on the area of the stream at EF, and not at 1F all on that at CD, and the latter may be made as small as we please without affecting the amount of water discharged. There is, however, a limit to this. .As the velocity at CD is greater than at EF the pressure is less, and therefore less than atmospheric pressure, if the discharge is into the air. If CD is so contracted that p=o, the continuity of flow is impossible. In fact the stream disengages itself from the mouthpiece for some value of p greater than o (fig. 61). From the equations, p/G =pa/G — (v2 —v 2)/2g• Let S2/w=m. Then v=vlm; p/G=pa(G—vi2(m2—1)(2g =pa/G—(m2—)h; whence we find that p/G will become zero or negative if 52/w-V (h+pd/G)/h} (1+pa./Gh}; or, putting p3/G=34 ft., if In practice there will be an interruption of the full bore flow with a less ratio of ti/w, because of the disengagement of air from the water. But, supposing this does not occur, the maximum discharge of a mouthpiece of this kind is Q =See also:call (2g(h+pa/G)} ; that is, the discharge is the same as for a well-bellmouthed mouth-piece of area w, and without the expanding part, discharging into a vacuum. § 52. Jet Pump.—A divergent mouthpiece may be arranged to See also:act as a pump, as shown in fig. 62. The water which supplies the energy ),A - [DISCHARGE OF ORIFICES required for pumping enters at A. The water to be pumped enters at B. The streams combine at DD where the velocity is greatest and the pressure least. Beyond DD the stream enlarges in section, and its pressure increases, till it is sufficient to See also:balance the head due to the height of the lift, and the water flows away ay the discharge pipe C. Additional information and CommentsThere are no comments yet for this article.
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