Then_Encyclopaedia Britannica, 11th Edition, "Husband" to "Hydrolysis" Volume 14, Slice 1

c_v = v_a [root](2gh) = [root](x²/4yh).

In the case of large orifices such as weirs, the velocity can be

directly determined by using a Pitot tube (§ 144).

[Illustration: FIG. 17.]

The coefficient of discharge, which for practical purposes is the most

important of the three coefficients, is best determined by tank

measurement of the flow from the given orifice in a suitable time. If

Q is the discharge measured in the tank per second, then

c = Q/[omega][root](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.

[Illustration: FIG. 18.]

For well made sharp-edged 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 h1, h2 be the heads, [omega]1, [omega]2 the areas of the orifices,

c1, c2 the coefficients. Then since the flow through each orifice is

the same

Q = c1[omega]1 [root](2gh1) = c2[omega]2 [root](2gh2).

c2 = c1([omega]1/[omega]2) [root](h1/h2).

[Illustration: FIG. 19.]

§ 19. _Coefficients for Bellmouths and Bellmouthed Orifices._--If an

orifice is furnished with a mouthpiece exactly of the form of the

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, c_c 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 bell-mouthed

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_v | .959 | .967 | .975 | .994 | .994 |

| Coefficient of resistance = c_r| .087 | .069 | .052 | .012 | .012 |

+--------------------------------+------+------+------+------+-------+

As there is no contraction after the jet issues from the orifice, c_c

= 1, c = c_v; and therefore

Q = c(v)[omega][root](2gh) = [omega][root]{2gh/(1 + c_r}.

§ 20. _Coefficients for Sharp-edged or virtually Sharp-edged

Orifices._--There are a very large number of measurements of discharge

from sharp-edged 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, Sharp-edged,

with free Discharge into the Air._ Q = c[omega][root](2gh).

+-----------+------------------------------------------------+

| Head | Diameters of Orifice. |

|measured to+------+------+------+------+------+------+------+

| Centre of | .02 | .04 | .10 | .20 | .40 | .60 | 1.0 |

| Orifice. +------+------+------+------+------+------+------+

| | Values of C. |

+-----------+------+------+------+------+------+------+------+

| 0.3 | .. | .. | .621 | .. | .. | .. | .. |

| 0.4 | .. | .637 | .618 | .. | .. | .. | .. |

| 0.6 | .655 | .630 | .613 | .601 | .596 | .588 | .. |

| 0.8 | .648 | .626 | .610 | .601 | .597 | .594 | .583 |

| 1.0 | .644 | .623 | .608 | .600 | .598 | .595 | .591 |

| 2.0 | .632 | .614 | .604 | .599 | .599 | .597 | .595 |

| 4.0 | .623 | .609 | .602 | .599 | .598 | .597 | .596 |

| 8.0 | .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 sharp-edged 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 Sharp-edged Orifices._

+----+------------------------------------------------------------------+

| | Form of Orifice. |

| +------+----------------+-----------------+-----------------+------+

| in | Cir- +------+---------+---------+-------+---------+-------+ Tri- |

+----+------+------+---------+---------+-------+---------+-------+------+

| 1 | .620 | .627 | .628 | .642 | .643 | .663 | .664 | .636 |

| 2 | .613 | .620 | .628 | .634 | .636 | .650 | .651 | .628 |

| 4 | .608 | .616 | .618 | .628 | .629 | .641 | .642 | .623 |

| 6 | .607 | .614 | .616 | .626 | .627 | .637 | .637 | .620 |

| 8 | .606 | .613 | .614 | .623 | .625 | .634 | .635 | .619 |

| 10 | .605 | .612 | .613 | .622 | .624 | .632 | .633 | .618 |

| 12 | .604 | .611 | .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 | .620 | .621 | .630 | .629 | .616 |

| 20 | .603 | .609 | .611 | .620 | .621 | .629 | .628 | .616 |

+----+------+------+---------+---------+-------+---------+-------+------+

The orifice was 0.196 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 (_Proc. 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 = 0.6075 + 0.0098/[root]h - 0.0037d,

where h is in feet and d in inches.

_Coefficients of Discharge from Circular Orifices. Temperature 51° to

55°._

+-------+--------------------------------------------------------------+

|Head in| Diameters of Orifices in Inches (d). |

| feet +------+------+------+------+------+------+------+------+------+

| h. | 1 | 1¼ | 1½ | 1¾ | 2 | 2¼ | 2½ | 2¾ | 3 |

+-------+------+------+------+------+------+------+------+------+------+

| | Coefficients (c). |

| +------+------+------+------+------+------+------+------+------+

| .75 | .616 | .614 | .616 | .610 | .616 | .612 | .607 | .607 | .609 |

| 1.0 | .613 | .612 | .612 | .611 | .612 | .611 | .604 | .608 | .609 |

| 1.25 | .613 | .614 | .610 | .608 | .612 | .608 | .605 | .605 | .606 |

| 1.50 | .610 | .612 | .611 | .606 | .610 | .607 | .603 | .607 | .605 |

| 1.75 | .612 | .611 | .611 | .605 | .611 | .605 | .604 | .607 | .605 |

| 2.00 | .609 | .613 | .609 | .606 | .609 | .606 | .604 | .604 | .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 still-water

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, Sharp-edged, in

Vertical Plane Surfaces._

+--------+----------------------------------------------------------------+

| Head | Ratio of Height to Width. |

| to | |

| Centre +------+------+------+------+--------+--------+--------+---------+

| of | | | | | | | | |

|Orifice.| 4 | 2 | 1½ | 1 | ¾ | ½ | ¼ | 1/8 |

+--------+------+------+------+------+--------+--------+--------+---------+

| | 4 ft.| 2 ft.|1½ ft.| 1 ft.|0.75 ft.|0.50 ft.|0.25 ft.|0.125 ft.|

| | 1 ft.| 1 ft.| 1 ft.| 1 ft.| 1 ft. | 1 ft. | 1 ft. | 1 ft. |

+--------+------+------+------+------+--------+--------+--------+---------+

| 0.2 | .. | .. | .. | .. | .. | .. | .. | .6333 |

| .3 | .. | .. | .. | .. | .. | .. | .6293 | .6334 |

| .4 | .. | .. | .. | .. | .. | .6140 | .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 | .6086 | .6168 | .6323 | .6324 |

| 1.0 | .. | .. |.6135 |.6010 | .6090 | .6172 | .6320 | .6320 |

| 1.25 | .. |.6188 |.6140 |.6018 | .6095 | .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 | .. |.6180 |.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 |.6160 |.6123 |.6050 | .6094 | .6146 | .6254 | .6254 |

| 4 |.6245 |.6150 |.6110 |.6047 | .6085 | .6136 | .6236 | .6236 |

| 4.5 |.6226 |.6138 |.6100 |.6044 | .6074 | .6125 | .6222 | .6222 |

| 5 |.6208 |.6124 |.6088 |.6038 | .6063 | .6114 | .6202 | .6202 |

| 6 |.6158 |.6094 |.6063 |.6020 | .6044 | .6087 | .6154 | .6154 |

| 7 |.6124 |.6064 |.6038 |.6011 | .6032 | .6058 | .6110 | .6114 |

| 8 |.6090 |.6036 |.6022 |.6010 | .6022 | .6033 | .6073 | .6087 |

| 9 |.6060 |.6020 |.6014 |.6010 | .6015 | .6020 | .6045 | .6070 |

| 10 |.6035 |.6015 |.6010 |.6010 | .6010 | .6010 | .6030 | .6060 |

| 15 |.6040 |.6018 |.6010 |.6011 | .6012 | .6013 | .6033 | .6066 |

| 20 |.6045 |.6024 |.6012 |.6012 | .6014 | .6018 | .6036 | .6074 |

| 25 |.6048 |.6028 |.6014 |.6012 | .6016 | .6022 | .6040 | .6083 |

| 30 |.6054 |.6034 |.6017 |.6013 | .6018 | .6027 | .6044 | .6092 |

| 35 |.6060 |.6039 |.6021 |.6014 | .6022 | .6032 | .6049 | .6103 |

| 40 |.6066 |.6045 |.6025 |.6015 | .6026 | .6037 | .6055 | .6114 |

| 45 |.6054 |.6052 |.6029 |.6016 | .6030 | .6043 | .6062 | .6125 |

| 50 |.6086 |.6060 |.6034 |.6018 | .6035 | .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 arrangements 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(H - h) [root]{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 | |

|above | Height of Orifice, H - h, in feet |

|upper +-----------------------+-----------------------+-----------------------+-----------------------+

|edge of | 1.31 | 0.66 | 0.16 | 0.10 |

|Orifice +-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+

|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 | 0.666 | 0.710 | 0.694 | 0.696 |

| .656 | 0.609 | 0.653 | 0.657 | 0.640 | 0.672 | 0.675 | 0.685 | 0.687 | 0.688 | 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 | 0.656 | 0.660 | 0.641 | 0.675 | 0.678 | 0.683 | 0.693 | 0.695 | 0.692 | 0.709 | 0.711 |

| 1.968 | 0.618 | 0.649 | 0.653 | 0.640 | 0.676 | 0.679 | 0.678 | 0.695 | 0.697 | 0.688 | 0.710 | 0.712 |

| 3.28 | 0.608 | 0.632 | 0.634 | 0.638 | 0.674 | 0.676 | 0.673 | 0.694 | 0.695 | 0.680 | 0.704 | 0.705 |

| 4.27 | 0.602 | 0.624 | 0.626 | 0.637 | 0.673 | 0.675 | 0.672 | 0.693 | 0.694 | 0.678 | 0.701 | 0.702 |

| 4.92 | 0.598 | 0.620 | 0.622 | 0.637 | 0.673 | 0.674 | 0.672 | 0.692 | 0.693 | 0.676 | 0.699 | 0.699 |

| 5.58 | 0.596 | 0.618 | 0.620 | 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 | 0.592 | 0.611 | 0.612 | 0.634 | 0.669 | 0.670 | 0.668 | 0.689 | 0.690 | 0.672 | 0.693 | 0.693 |

+--------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+

For rectangular orifices,

C_c = 0.62(1 + 0.152n/p);

and for circular orifices,

C_c = 0.62(1 + 0.128n/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 c_c, when the border extends

over ¼, ½, or ¾ of the whole perimeter:--

+--------+-----------------------+--------------------+

| | C_c | C_c |

| n/p | Rectangular Orifices. | Circular Orifices. |

+--------+-----------------------+--------------------+

| 0.25 | 0.643 | .640 |

| 0.50 | 0.667 | .660 |

| 0.75 | 0.691 | .680 |

+--------+-----------------------+--------------------+

[Illustration: FIG. 20.]

[Illustration: FIG. 21.]

For larger values of n/p 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 2.7 times the corresponding width of the orifice. The

coefficients of contraction for this case are imperfectly known.

[Illustration: FIG. 22.]

§ 24. _Orifices Furnished with Channels of Discharge._--These external

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).

+-----------+-------------------------------------------------------+

| h2--h1 | h1 in feet. |

| in feet |------+-----+-----+-----+------+-----+-----+-----+-----+

| |.0656 |.164 |.328 |.656 |1.640 |3.28 |4.92 |6.56 |9.84 |

+-----------+------+-----+-----+-----+------+-----+-----+-----+-----+

| A\ | .480 |.511 |.542 |.574 | .599 |.601 |.601 |.601 |.601 |

| B > 0.656 | .480 |.510 |.538 |.506 | .592 |.600 |.602 |.602 |.601 |

| C/ | .527 |.553 |.574 |.592 | .607 |.610 |.610 |.609 |.608 |

| | | | | | | | | | |

| A\ | .488 |.577 |.624 |.631 | .625 |.624 |.619 |.613 |.606 |

| B > 0.164 | .487 |.571 |.606 |.617 | .626 |.628 |.627 |.623 |.618 |

| C/ | .585 |.614 |.633 |.645 | .652 |.651 |.650 |.650 |.649 |

+-----------+------+-----+-----+-----+------+-----+-----+-----+-----+

[Illustration: FIG. 23.]

§ 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

marked peculiarity of form. But if the orifice is in a vertical

surface, and if its dimensions are not small compared with the head,

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.

[Illustration: FIG. 24.]

Let a1, a2 (fig. 24) be two points in an orifice at depths h1, h2 from

the free surface. The filaments issuing at a1, a2 will have the

different velocities [root](2gh1) and [root](2gh2). Consequently they

will tend to describe parabolic paths a1cb1 and a2cb2 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

VV. C. Unwin found (_Phil. Mag._, October 1878, p. 281) that for

sharp-edged orifices temperature has a very small influence on the

discharge. For an orifice 1 cm. in diameter with heads of about 1 to

1½ ft. the coefficients were:--

Temperature F. C.

205° .594

62° .598

For a conoidal or bell-mouthed orifice 1 cm. diameter the effect of

temperature was greater:--

Temperature F. C.

190° 0.987

130° 0.974

60° 0.942

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 sharp-edged orifice 2½ 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 0.98l at 170° F. The corresponding

coefficients of resistance are 0.0828 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 ¾-in.

nozzle; 0.982 for 7/8 in.; 0.972 for 1 in.; 0.976 for 1(1/8) in.;

and 0.971 for 1¼ in. The nozzles were fixed on a taper play-pipe, 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.

Encyclopaedia Britannica, 11th Edition, "Husband" to "Hydrolysis" Volume 14, Slice 1 · 第 41 / 207 题