DISCHARGE OF LIQUIDS]
The relation between c, and cr for any orifice is easily found:
v„= cvJ2gh = J 12ygh/(1 +cr)]
cv=J {I/(1Fcr)). (5)
cr= I/c,2—I. (5a) Thus if ci, =o.97, then cr=o•o628, That is, for such an orifice about 61% of the head is expended in overcoming frictional resistances to flow.
Coefficient of Contraction—Sharpedged Orifices in Plane Surfaces.—When a jet issues from an aperture in a vessel, it may either spring
clear from the inner edge of the orifice as at a or b (fig. 15), or it may adhere to the sides of the orifice as at c. The former condition will be found if the orifice is bevelled outwards as at a, so as to be sharp edged, and it will also occur generally for a prismatic aperture like b, provided the thickness of the plate in which the aperture is formed is less than the diameter
of the jet. But if the thickness is greater the condition shown at c will occur.
When the discharge occurs as at a or b, the filaments converging towards the orifice continue to converge beyond it, so that the section of the jet where the filaments have become parallel is smaller than the section of the orifice. The inertia of the filaments opposes sudden change of direction of motion at the edge of the orifice, and the convergence continues for a distance of about half the diameter of the orifice beyond it. Let w be the
area of the orifice, and c,w the area of the jet at the point where convergence ceases; then cc is a coefficient to be determined experimentally for each kind of orifice, called the coefficient of contraction. When the orifice is a sharpedged orifice in a plane surface, the value of c~ is on the average 0.64, or the section of the jet is very nearly fiveeighths of the area of the orifice.
Coefficient of Discharge.—In applying the general formula Q=wv to a stream, it is assumed that the filaments have a common velocity v normal to the section w. But if the jet contracts, it is at the contracted section of the jet that the direction of motion is normal to a transverse section of the jct. Hence the actual discharge when contraction occurs is
Qa = c,,v X co= ccc„wJ (2gh), or simply, if c=c„cc,
Qa = cwsl (2gh),
where c is called the coefficient of discharge. Thus for a sharpedged plane orifice c=0•97X 0.64 =0.62.
§ 18. Experimental Determination of c,,, c~, and c.—The coefficient of contraction cc is directly determined by measuring the dimensions of the jet.
For this purpose fixed screws of fine pitch (fig. 16) are convenient. These are set to touch the jet, and then the distance between them can be measured at leisure.
The coefficient of velocity is determined directly by measuring the parabolic path of a horizontal jet.
Let OX, OY (fig. 17) be horizontal and vertical axes, the origin being at the orifice. Let h be the head, and x, y the coordinates of a point A on the parabolic path of the jet. If va is the velocity at39
the orifice, and t the time in which a particle moves from 0 to A, then
x=vat ; Y=zgt2•
va = J (gx2/2y).
cr =va/J (2gh) =J (x2/4Yh).
In the case of large orifices such as weirs, the velocity can be directly determined by using a Pitot tube (§ 144).
The coefficient of discharge, which for practical purposes is the most important of the three coefficients, is best determined by tank
measurement of r
the flow from the
given orifice in a =_= T=
suitable time. If —_
Q is the discharge A
measured in the tank per second, then
c=Q/wJ (2gh). Measurements of this kind though simple in principle are not free from some practical difficulties, and require much care. In fig. 18 is shown an arrangement of measuring tank. The orifice is fixed
in the wall of the cistern A and discharges either into the waste channel BB, or into the measuring tank. There is a short trough on rollers C which when run under the jet directs the discharge into the tank, and when run back again allows the discharge to drop
into the waste channel. D is a stilling screen to prevent agitation of the surface at the measuring point, E, and F is a discharge valve for emptying the measuring tank. The rise of level in the tank, the time of the flow and the head over the orifice at that time must be exactly observed.
For well made sharpedged orifices, small relatively to the water surface in the supply reservoir, the coefficients under different conditions of head are pretty exactly known. Suppose the same quantity of water is made to flow in succession through such an orifice and through another orifice of which the coefficient is required, and when the rate of flow is constant the heads over each orifice are noted. Let hi, h2 be the heads, wi, w2 the areas of the orifices, c,, c2 the coefficients. Then since the flow through each orifice is the same
Q =c,wis/ (2ghi) =c2w2J (2gh2).
c2 = ct(0.7i/w2) J (hi/h2).
§ 19. Coefficients for Bellmouths and Bellmouthed Orifices.—If an orifice is furnished with a mouthpiece exactly of the form of the
D=1•asd
liil~ ij6i91 iI
Eliminating t, Then
e,..._
~: II ~,~: k
list'
A
C
at=0.8D. "i
contracted vein, then the whole of the contraction occurs within the mouthpiece, and if the area of the orifice is measured at the smaller end, cr must be put =1. It is often desirable to bellmouth the ends of pipes, to avoid the loss of head which occurs if this is
not done; and such a bellmouth may also have the form of the contracted jet. Fig. 19 shows the proportions of such a bellmouth or bellmouthed orifice, which approximates to the form of the contracted jet sufficiently for any practical purpose.
For such an orifice L. J. Weisbach found the following values of the coefficients with different heads.
Head over orifice, in ft. = h •66 1.64 11.48 55.77 337.93
Coefficient of velocity=c . '959 '967 '975 '994 '994
i Coefficient of resistance=cr •o87 •069 •052 •012 •012
As there is no contraction after the jet issues from the orifice, c=1, c=c,,; and therefore
Q =c,,(0\1 (2gh) =ceJ {2gh/(1 +cr)}.
§ 2o. Coefficients for Sharpedged or virtually Sharpedged Orifices.There are a very large number of measurements of discharge from sharpedged orifices under different conditions of head. An account of these and a very careful tabulation of the average values of the coefficients will be found in the Hydraulics of the late Hamilton Smith (Wiley & Sons, New York, 1886). The following short table abstracted from a larger one will give a fair notion of how the coefficient varies according to the most trustworthy of the experiments.
Coefficient of Discharge for Vertical Circular Orifices, Sharpedged,
with free Discharge into the Air. Q=cwJ (2gh).
Head Diameters of Orifice.
measured to •02 •04 •10 •20 •40 •60 1.0
Centre of
Orifice. Values of C.
0.3 .. 6z1
0.4 .637 •618 .
o•6 •655 •630 •613 •6o1 •596 •588
0.8 •648 •626 •610 •6o1 •597 '594 '583
1•o .644 .623 '6o8 .60o 598 '595 '591
2•o .632 •614 •604 •599 '599 .597 .595
4 0 •623 .609 •602 •599 •598 '597 '596
8•o .614 •605 •600 •598 .597 '596 '596
20.0 •601 '599 '596 '596 '596 '596 '594
At the same time it must be observed that differences of sharpness in the edge of the orifice and some other circumstances affect the results, so that the values found by different careful experimenters are not a little discrepant. When exact measurement of flow has to be made by a sharpedged orifice it is desirable that the coefficient for the particular orifice should be directly determined.
The following results were obtained by Dr H. T. Bovey in the laboratory of McGill University.
Coefficient of Discharge for Sharpedged Orifices.
Form of Orifice.
Square. Rectangular Ratio Rectangular Ratio
of Sides 4:1. of Sides 16:r.
Head in •
ft. Cir Lon Tri
cular. Sides Dia Long I•ong Long Sides angular.
vertical gonal Sides Sides Sides hori
vertical. vertical. horzontail. vertical. zontal.
r •62o •627 •628 •642 .643 •663 •664 .636
2 •613 •62o '628 •634 •636 •65o .65t •628
4 •6o8 •616 •618 •628 .629 •641 •642 •623
6 .607 '614 •616 •626 •627 '637 '637 '620
8 •6o6 •613 .614 '623 .625 '634 '635 •619
10 •605 •612 •613 •622 •624 .632 .633 •618
12 •604 •6,1 •612 •622 •623 •631 •631 •618
14 604 •610 •612 •621 •622 •630 •630 •618
16 .603 •610 •611 •620 •622 •630 .630 •617
18 .603 •610 •611 •62o •621 .630 •629 •616
20 •603 .609 •611 •620 •621 •629 •628 •616
The orifice was o. 196 sq. in. area and the reductions were made with g=32.176 the value for Montreal. The value of the coefficient appears to increase as (perimeter) / (area) increases. It decreases as the head increases. It decreases a little as the size of the orifice is greater.
Very careful experiments by J. G. Mair (Prot. Inst. Civ. Eng. lxxxiv.) on the discharge from circular orifices gave the results shown on top of next column.
The edges of the orifices were got up with scrapers to a sharp square edge. The coefficients generally fall as the head increases and as the diameter increases. Professor W. C. Unwin found that the results agree with the formula
c = 0.607 5 i o • / J h  0.003 7d,
where h is in feet and d in inches.Coefficients of Discharge from Circular Orifices. Temperature 51 ° to 55°.
1 Head in Diameters of Orifices in Inches (d).
feet
h.
I 14 11 I I4 2 24 2z 24 13
Coefficients (c).
•75 •616 •614 •616 •6,0 •616 •612 •607 •607 •609
1•o •613 •612 •612 •611 •612 •611 •604 •6o8 •609
1.25 .613 •614 •610 •6o8 •612 •6o8 •6o5 .6o5 •6o6
1.50 •610 •612 •611 •6o6 •610 •607 •603 •607 •6o5
1.75 •612 •611 •611 •6o5 •611 .605 •604 •607 •6o5
2.00 •609 •613 •609 •6o6 •609 •6o6 .6o¢ .6o¢ •605
The following table, compiled by J. T. Fanning (Treatise on Water Supply Engineering), gives values for rectangular orifices in vertical plane surfaces, the head being measured, not immediately over the orifice, where the surface is depressed, but to the stillwater surface at some distance from the orifice. The values were obtained by graphic interpolation, all the most reliable experiments being plotted and curves drawn so as to average the discrepancies.
Coefficients of Discharge for Rectangular Orifices, Sharpedged,
in Vertical Plane Surfaces.
Head to Ratio of Height to Width.
Centre of 4 2 }
Orifice.
11 r i I
ai ,a ° m a., yy ai C 'w A . m
440.0 ee ;O eo;C ai mat m ai
4 4.1 .. 751.
Feet. •3 . 3 .c 3 «•3 ~,

e N .. .... .... o ow o .t1 w
o..
0.2 .. .. .. .. .. .. .. •6333
•3 •6293 •6334
'4 .. .. .. .. . 614o •6306 •6334
.5 .. .. .. .. •6050 •6150 .6313 •6333
6 .. .. '5984 •6063 •6156 •6317 •6332
'7 •• •• •. '5994 •6074 •6162. •6319 •6328
•8 .. .. •6130 •6000 •6082 .6165 .6322 •6326
•9 .. .. '6134 •6006 •6o86 •6168 .6323 '6324
1•o .. •6135 •6oio •6090 .6172 .6320 •6320
P25 . . •6188 •6140 •6o18 •6o95 •6173 .6317 '6312
1.50 .. •6187 •6144 •6026 •6100 •6172 •6313 •6303
1.75 .. •6186 •6145 •6033 •6103 •6168 •6307 •6296
2 .. •6183 •6144 •6036 .6104 •6166 •6302 •6291
2.25 •618o •6143 •6029 •6103 •6163 .6293 •6286
2.50 •6290 •6176 '6139 •6043 •6102 •6157 •6282 •6278
2'75 •6280 •6173 •6136 •6046 •6101 •6155 •6274 •6273
3 '6273 •6170 .6132 .6048 •6100 •6153 •6267 •6267
3.5 •6250 •616o •6123 •6050 '6094 •6146 .6254 •6254
4 '6245 •6150 •6110 .6047 •6o85 •6136 •6236 •6236
4'5 •6226 •6138 •6100 .6044 •6074 •6125 •6222 •6222
5 •6208 •6124 '6o88 •6038 •6063 •6114 •62o2 •6202
6 •6158 •6094 •6063 •6020 •6044 •6087 •6154 •6154
7 •6124 .6064 •6038 •6ot1 •6o32 •6058 .6110 •6114
8 .6090 .6036 •6o22 •60,0 •6022 •6033 .6073 •6o87
9 •6o6o •6020 •6014 •6oto •6015 •6020 .6045 •6070
to •6035 .6015 •6oto •60,0 •6o,o •6o,o •6030 •6o6o
15 •6o4o •6o,8 •60,0 •6oit •6012 •6013 •6033 •6o66
20 •6045 .6024 •6012 •6012 •6014 •6018 .6036 •6074
25 •6048 •6o28 •6014 •6012 •6o,6 •6022 •6040 '6083
30 •6054 .6034 •6017 •6013 '6o,8 .6027 •6044 •6o92
.35 •6o6o •6039 •6021 .6014 •6o22 •6o32 •6049 .6103
40 •6o66 .6045 '6025 •6015 •6026 '6037 .6055 •6114
45 •6o54 •6o52 •6029 .6o,6 •6o3o •6043 •6o62 •6125
50 •6o86 •6o6o .6034 •6o18 •6o35 .6050 .6070 •6140
§ 21. Orifices with Edges of Sensible Thickness.When the edges of the orifice are not bevelled outwards, but have a sensible thickness, the coefficient of discharge is somewhat altered. The following table gives values of the coefficient of discharge for the arrangerrieats of the orifice shown in vertical section at P, Q, R (fig. 20). The plan of all the orifices is shown at S. The planks. forming the orifice and sluice were each 2 in. thick, and the orifices were all 24 in. wide. The heads were measured immediately over the orifice. In this case,
Q=cb(Hh)J {2g(H+h)/2}.
§ 22. Partially Suppressed Contraction.Since the contraction of the jet is due to the convergence towards the orifice of the issuing streams, it will be diminished if for any portion of the edge of the orifice the convergence is prevented. Thus, if an internal rim or border is applied to part of the edge of the orifice (fig. 21), the convergence for so much of the edge is suppressed. For such cases G. Bidone found the following empirical formulae applicable:
Table of Coefficients of Discharge for Rectangular Vertical Orifices in Fig. 20.
Head h Height of Orifice, H h, in feet.
above
upper 1.31 o•66 0.16 0.10
edg  of
Orifi e in feet. P Q R P Q R P Q R P Q R
0.328 0.598 0.644 0.648 0.634 0.665 0.668 0.691 0.664 o•666 0.710 0.694 0.696
.656 0.609 0.653 0.657 0.640 0.672 0.675 0.685 0.687 o•688 0.696 0.704 0.706
•787 0.612 0.655 0.659 0.641 0.674 0.677 0.684 0.690 0.692 0.694 0.706 0.708
.984 0.616 o•656 o•660 0.641 0.675 0.678 0.683 0.693 0.695 0.692 0.709 0.711
1.968 o•618 0.649 0.653 0.640 0.676 0.679 o•678 0.695 0.697 o•688 0.710 0.712
3.28 o•6o8 0.632 0.634 0'638 0'674 0.676 0.673 0.694 0.695 0.68o ti 0.704 0.705
4.27 0.602 0.624 0.626 0.637 0'673 0.675 I 0.672 0.693 0.694 0.678 0.701 0.702
4'92 0'598 l 0.620 0.622 0.637 I 0'673 0.674 0.672 0.692 0.693 0.676 0.699 0.699
5.58 0.596 o•618 0.62o 0.637 0.672 0.673 0.672 0.692 0.693 0.676 0.698 0.698
6.56 0.595 0.615 0.617 0.636 0.671 0.672 0.671 0.691 0.692 0.675 0.696 0.696
9.84 0•J92 o.611 0.612 0.634 0.669 0.670 0.668 0.689 0.690 0.672 0.693 0.693
For rectangular orifices,
e~ = 0.62(1+0.152707) ;
and for circular orifices,
c , =0.62(1 +o• 128n/p) ;
when n is the length of the edge of the orifice over which the border extends, and p is the whole length of edge or perimeter of the orifice. The following are the values of cc, when the border extends over
4 or ; of the whole perimeter:
c, Circular Orifices.
Rectangular Orifices.
0'643 •64o
0.667 •66o
0.691 .68o
For larger values of nip
§ 24. Orifices Furnished with Channels of Discharge.These ex
ternal borders to an orifice also modify the contraction.
The following coefficients of discharge were obtained with openings 8 in. wide, and small in proportion to the channel of approach (fig. 22, A, B, C).
h, in feet.
k_,feeth, in
. I
'0656 '164 '328 •656 3'28 4:92 6'56 9'84
A •48o 511 '542 '599 •6oi •60I •6oi •6oi
B 0.656 •48o •510 '538 •506 .592 •600 •602 •602 •6oi
C .527 '553 '574 .592 •607 •6io •6io •609 •6o8
A ( '48 '577 .624 .631 .625 •624 .619 •613 •6o6
B o•I64 487 .571 .6o6 •617 •626 •628 .627 .623 •618
)
C '585 •614 .633 .645 .652 •651 •650 .65o '649
§ 25. Inversion of the Jet.When a jet issues from a horizontal orifice, or is of small size compared with the head, it presents no
b
n/ p
0.25 0.50 0.75
the formulae are not applicable. C. R. Bornemann has shown, however, that these formulae for suppressed contraction are not reliable.
§ 23. Imperfect Contraction.If the sides of the vessel approach near to the edge of the orifice, they interfere with the convergence of the streams to which the contraction is due, and the contraction is then modified. It is generally stated that the influence of the sides begins to be felt if their distance from the edge of the orifice is less than 27 times the corresponding
width of the orifice. The coefficients of contraction for this case j marked peculiarity of form. But if the orifice is in a vertical surare imperfectly known. ( face, and if its dimensions are not small compared with the head,
C
Slope I in 20
1
A
h, h,
10
B
it undergoes a series of singular changes of form after leaving the orifice. These were first investigated by G. Bidone (1781—1839); subsequently H. G. Magnus (1802—1870) measured jets from different orifices; and later Lord Rayleigh (Proc. Roy. Soc. xxix. 71) investigated them anew.
Fig. 23 shows some forms, the upper figure giving the shape of the orifices, and the others sections of the jet. The jet first contracts as described above, in consequence of the convergence of the fluid streams within the vessel, retaining, however, a form similar to that of the orifice. Afterwards it expands into sheets in planes perpendicular to the sides of the orifice. Thus the jet from a triangular orifice expands into three sheets, in planes bisecting at right angles the three sides of the triangle. Generally a jet from an orifice, in the form of a regular polygon of n sides, forms n sheets in planes perpendicular to the sides of the polygon.
Bidone explains this by reference to the simpler case of meeting streams. If two equal streams having the same axis, but moving in opposite directions, meet, they spread out into a thin disk normal to the common axis of the streams. If the directions of two streams intersect obliquely they spread into a symmetrical sheet perpendicular to the plane of the streams.
Let al, a2 (fig. 24) be two points in an orifice at depths hr, h2 from the free surface. The filaments issuing at al, a2 will have the different velocities d 2ghi and if 2gh2. Consequently they will tend to describe parabolic paths click and ascbs of different horizontal range, and intersecting in the point c. But since two filaments cannot simultaneously flow through the same point, they must exercise mutual pressure, and will be deflected out of the paths they tend to describe. It is this mutual pressure which causes the expansion of the jet into sheets.
Lord Rayleigh pointed out that, when the orifices are small and the head is not great, the expansion of the sheets in directions perpendicular to the direction of flow reaches a limit. Sections taken at greater distance from the orifice show a contraction of the sheets until a compact form is reached similar to that at the first contraction. Beyond this point, if the jet retains its coherence, sheets are thrown out again, but in directions bisecting the angles between the previous sheets. Lord Rayleigh accepts an explanation of this contraction first suggested by H. Buff (1805—1878), namely, that it is due to surface tension.
§ 26. Influence of Temperature on Discharge of Orifices.—Professor W. C. Unwin found (Phil. Mag., October 1878, p. 281) that for sharpedged orifices temperature has a very small influence on the discharge. For an orifice i cm. in diameter with heads of about i to I ft. the coefficients were:
Temperature F. .
205°°
6
2
For a conoidal or bellmouthed orifice i cm. diameter the effect of temperature was greater:
Temperature F. C.
190° 0.987
130°  0.974
° 94
60 o• 2
an increase in velocity of discharge of 4% when the temperature increased 130°.
J. G. Mair repeated these experiments on a much larger scale (Proc. Inst. Civ. Eng. lxxxiv.). For a sharpedged orifice 21 in. diameter, with a head of 1.75 ft., the coefficient was 0.604 at 57° and 0.607 at 179° F., a very small difference. With a conoidal orifice the coefficient was 0.961 at 55° and 0.981 at 170° F. The corresponding coefficients of resistance are o•o828 and 0.0391, showing that the resistance decreases to about half at the higher temperature.
§ 27. Fire Hose Nozzles. Experiments have been made by J. R. Freeman on the coefficient of discharge from smooth cone nozzles used for fire purposes. The coefficient was found to be 0.983 for 1in. nozzle; 0.982 fore in.; 0.972 for 1 in.; 0.976 for 18 in.; and 0.971 for 11 in. The nozzles were fixed on a taper playpipe, and the coefficient includes the resistance of this pipe (Amer. Soc. Civ. Eng. xxi.. 1889). Other forms of nozzle were tried such as ring nozzles for which the coefficient was smaller.
IV. THEORY OF THE STEADY MOTION OF FLUIDS.
§ 28. The general equation of the steady motion of a fluid given under Hydrodynamics furnishes immediately three results as to the distribution of pressure in a stream which may here be assumed.
(a) If the motion is rectilinear and uniform, the variation of pressure is the same as in a fluid at rest. In a stream flowing in anopen channel, for instance, when the effect of eddies produced by the roughness of the sides is neglected, the pressure at each point is simply the hydrostatic pressure due to the depth below the free surface.
(b) If the velocity of the fluid is very small, the distribution of pressure is approximately the same as in a fluid at rest.
(c) If the fluid molecules take precisely the accelerations which they would have if independent and submitted only to the external forces, the pressure is uniform. Thus in a jet falling freely in the air the pressure throughout any cross section is uniform and equal to the atmospheric pressure.
(d) In any bounded plane section traversed normally by streams which are rectilinear for a certain distance on either side of the section, the distribution of pressure is the same as in a fluid at rest.
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