SPECIAL CASES OF DISCHARGE FROM ORIFICES
§ 45. 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 has 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 hi, h2 (fig. 48) be the heads measured from the free surface to the 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)ihi]. (2)
In cases where u can be directly determined, these formulae 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 12 be the sectional area of the channel where h, and 112 are measured. Then u=Q/ 12 and 1)=Q2/2g 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 equation
Q'= 3cbd 2g]hhi—h0].
Then I) = Q'2/2gl22 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 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 different conditions. A filament M,mi (fig. 49) in the upper 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 constant. If Qi, Q2 are the discharges from the upper and lower parts of the orifice, b the width of the orifice, then
Q1=JcbN/2glhi—hit]
Q2=cb(h2—h)d2gh
In the case of a rectangular notch or 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 crest
(fig. 5o).
From some experiments by Messrs A. Fteley and F. P. Stearns (Trans. Am. 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 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
value of c in this case is
oiiai~i i~~%_~~~i~~~Tiiis~/i~~i~•~~  imperfectly known. FIG. 50.
§ 47. Bazin's Researches on
Weirs.—H. Bazin has executed a long series of researches on the flow over weirs, so systematic and complete that they almost supersede other observations. The account of them is contained in a series of papers in the Annales des Ponts et Chaussees (October 1888, January 189o, November 1891, February 1894, December 1896, 2nd trimestre 1898). Only a very abbreviated account can be given here. The general plan of the experiments was to establish first the coefficients of discharge for a 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 Canal de Bourgogne, taking water from it through three sluices 0.3 X I •o metres. The water enters a 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 depth of i•6 metres. From
this extends a channel 200 metres in length with a slope of I mm. per metre. The channel is 2 metres wide with vertical sides. The channels were constructed of concrete rendered with cement. The water levels were taken in chambers constructed near the canal, by floats actuating an index on a dial. 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 plate 4 in. thick. It was sharpedged with free overfall. It was as wide as the canal so that end contractions were suppressed, and enlargements were formed below the crest to admit air under the water 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 suppressed in all cases. Assuming the general formula
Q=mlhd (2gh), (I)
hZ
. . Ma
i
i
I my
(3)
flow of the stream, which causes a difference of surfacelevel above and below the
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 close agreement.
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 original 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 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 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 undersurface 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 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 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 independent of the tail water level or influenced by it. In the former case the fall is followed by a rapid, terminating in a standing wave. In the latter case when the 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 Flat Crests.
The water sheet may 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 condition is that of a sharpedged weir and it is realized when the head is at least 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 sharpedged weir in similar con
ditions. Rounding the up
stream edge even to a small h extent modifies the dis
charge. If R is the 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{ (hhl)/h},
where as before ml is the coefficient for a sharpedged 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 watersupply from streams in high moorland districts, in which the
flow is extremely variable. The water is collected in large storage reservoirs, from which an uniform supply can be sent to the town. In
such cases it is desirable to separate the coloured water which comes down the streams in high floods from the purer water of ordinary flow. The latter is sent into the reservoirs; the former is allowed
(2)
Values of µ
Vertical weir . . . Inclination down stream
HYDRAULICS
to flow away down the original stream channel, or is stored in separate reservoirs and used as compensation water. To accomplish the separation of the flood and ordinary water, advantage is taken of the different horizontal range of the parabolic path of the water falling over a weir, as the depth on the weir and, consequently, the velocity change. Fig. 55 shows one of these separating weirs in the form in which they were first introduced on the Manchester Waterworks; fig. 56 a more modern weir of the same kind designed by Sir A. Binnie for the 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 waste channel.
It may be assumed, probably with accuracy enough for 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 jet will just pass over the orifice. Set off ad vertically
d
3 v h
and equal to Zg on any scale; af horizontally and equal to a J (gh). 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 short cylindrical pipe or mouthpiece of a length at least equal to 11 times its smallest transverse dimension, the stream, after contraction within the mouthpiece, expands to fill it and issues full bore, or without contraction, at the point of discharge. The discharge is found to be about onethird greater than that from a simple orifice of the same size. On the other hand, the energy of the fluid per unit of 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 action being something like that which occurs at an abrupt change of section.
Let fig. 58 represent a vessel discharging through a cylindrical mouthpiece at the depth h from the free surface, and let the 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 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 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) (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 contraction 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 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 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 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 outside 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 external end of the mouthpiece, and ccfl the section of the contracted jet beyond the mouthpiece.
Convergent Mouthpieces (Castel's Experiments).—Smallest diameter of
orifice =o.o5085 ft. Length of mouthpiece 2.6 Diameters.
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
=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/wV (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 =call (2g(h+pa/G)} ;
that is, the discharge is the same as for a wellbellmouthed mouthpiece of area w, and without the expanding part, discharging into a vacuum.
§ 52. Jet Pump.—A divergent mouthpiece may be arranged to 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 balance the head due to the height of the lift, and the water flows away ay the discharge pipe C.
End of Article: SPECIAL CASES OF DISCHARGE FROM 

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