Chapter V: Theory of the Discharge From Orifices and Mouthpieces

§ 37. _Minimum Coefficient of Contraction. Re-entrant Mouthpiece of
Borda._--In one special case the coefficient of contraction can be
determined theoretically, and, as it is the case where the convergence
of the streams approaching the orifice takes place through the
greatest possible angle, the coefficient thus determined is the
minimum coefficient.

Let fig. 39 represent a vessel with vertical sides, OO being the free
water surface, at which the pressure is p_a. Suppose the liquid issues
by a horizontal mouthpiece, which is re-entrant and of the greatest
length which permits the jet to spring clear from the inner end of the
orifice, without adhering to its sides. With such an orifice the
velocity near the points CD is negligible, and the pressure at those
points may be taken equal to the hydrostatic pressure due to the depth
from the free surface. Let [Omega] be the area of the mouthpiece AB,
[omega] that of the contracted jet aa Suppose that in a short time t,
the mass OOaa comes to the position O´O´ a´a´; the impulse of the
horizontal external forces acting on the mass during that time is
equal to the horizontal change of momentum.

The pressure on the side OC of the mass will be balanced by the
pressure on the opposite side OE, and so for all other portions of the
vertical surfaces of the mass, excepting the portion EF opposite the
mouthpiece and the surface AaaB of the jet. On EF the pressure is
simply the hydrostatic pressure due to the depth, that is, (p_a + Gh).
On the surface and section AaaB of the jet, the horizontal resultant
of the pressure is equal to the atmospheric pressure p_a acting on the
vertical projection AB of the jet; that is, the resultant pressure is
-p_a[Omega]. Hence the resultant horizontal force for the whole mass
OOaa is (p_a + Gh)[Omega] - p_a[Omega] = Gh[Omega]. Its impulse in the
time t is Gh[Omega]t. Since the motion is steady there is no change of
momentum between O´O´ and aa. The change of horizontal momentum is,
therefore, the difference of the horizontal momentum lost in the space
OOO´O´ and gained in the space aaa´a´. In the former space there is no
horizontal momentum.

The volume of the space aaa´a´ is [omega]vt; the mass of liquid in
that space is (G/g)[omega]vt; its momentum is (G/g)[omega]v²t.
Equating impulse to momentum gained,

Gh[Omega] = (G/g)[omega]v²t;

.: [omega]/[Omega] = gh/v²

But

v² = 2gh, and [omega]/[Omega] = c_c;

.: [omega]/[Omega] = ½ = c_c;

a result confirmed by experiment with mouthpieces of this kind. A
similar theoretical investigation is not possible for orifices in
plane surfaces, because the velocity along the sides of the vessel in
the neighbourhood of the orifice is not so small that it can be
neglected. The resultant horizontal pressure is therefore greater than
Gh[Omega], and the contraction is less. The experimental values of the
coefficient of discharge for a re-entrant mouthpiece are 0.5149
(Borda), 0.5547 (Bidone), 0.5324 (Weisbach), values which differ
little from the theoretical value, 0.5, given above.

§ 38. _Velocity of Filaments issuing in a Jet._--A jet is composed of
fluid filaments or elementary streams, which start into motion at some
point in the interior of the vessel from which the fluid is
discharged, and gradually acquire the velocity of the jet. Let Mm,
fig. 40 be such a filament, the point M being taken where the velocity
is insensibly small, and m at the most contracted section of the jet,
where the filaments have become parallel and exercise uniform mutual
pressure. Take the free surface AB for datum line, and let p1, v1, h1,
be the pressure, velocity and depth below datum at M; p, v, h, the
corresponding quantities at m. Then § 29, eq. (3a),

v1²/2g + p1/G - h1 = v²/2g + p/G - h (1)

But at M, since the velocity is insensible, the pressure is the
hydrostatic pressure due to the depth; that is v1 = 0, p1 = p_a + Gh1.
At m, p = p_a, the atmospheric pressure round the jet. Hence,
inserting these values,

0 + p_a/G + h1 - h1 = v²/2g + p_a/G - h;

v²/2g = h; (2)

or v = [root](2gh) = 8.025V [root]h. (2a)

That is, neglecting the viscosity of the fluid, the velocity of
filaments at the contracted section of the jet is simply the velocity
due to the difference of level of the free surface in the reservoir
and the orifice. If the orifice is small in dimensions compared with
h, the filaments will all have nearly the same velocity, and if h is
measured to the centre of the orifice, the equation above gives the
mean velocity of the jet.

_Case of a Submerged Orifice._--Let the orifice discharge below the
level of the tail water. Then using the notation shown in fig. 41, we
have at M, v1 = 0, p1 = Gh; + p_a at m, p = Gh3 + p_a. Inserting these
values in (3), § 29,

0 + h1 + p_a/G - h1 = v²/2g + h3 - h2 + p_a/G;

v²/2g = h2 - h3 = h, (3)

where h is the difference of level of the head and tail water, and may
be termed the _effective head_ producing flow.

_Case where the Pressures are different on the Free Surface and at the
Orifice._--Let the fluid flow from a vessel in which the pressure is
p0 into a vessel in which the pressure is p, fig. 42. The pressure p0
will produce the same effect as a layer of fluid of thickness p0/G
added to the head water; and the pressure p, will produce the same
effect as a layer of thickness p/G added to the tail water. Hence the
effective difference of level, or effective head producing flow, will
be

h = h0 + p0/G - p/G;

and the velocity of discharge will be

v = [root][2g {h0 + (p0 - p)/G}]. (4)

We may express this result by saying that differences of pressure at
the free surface and at the orifice are to be reckoned as part of the
effective head.

Hence in all cases thus far treated the velocity of the jet is the
velocity due to the effective head, and the discharge, allowing for
contraction of the jet, is

Q = c[omega]v = c[omega] [root](2gh), (5)

where [omega] is the area of the orifice, c[omega] the area of the
contracted section of the jet, and h the effective head measured to
the centre of the orifice. If h and [omega] are taken in feet, Q is in
cubic feet per second.

It is obvious, however, that this formula assumes that all the
filaments have sensibly the same velocity. That will be true for
horizontal orifices, and very approximately true in other cases, if
the dimensions of the orifice are not large compared with the head h.
In large orifices in say a vertical surface, the value of h is
different for different filaments, and then the velocity of different
filaments is not sensibly the same.

SIMPLE ORIFICES--HEAD CONSTANT

§ 39. _Large Rectangular Jets from Orifices in Vertical Plane
Surfaces._--Let an orifice in a vertical plane surface be so formed
that it produces a jet having a rectangular contracted section with
vertical and horizontal sides. Let b (fig. 43) be the breadth of the
jet, h1 and h2 the depths below the free surface of its upper and
lower surfaces. Consider a lamina of the jet between the depths h and
h + dh. Its normal section is bdh, and the velocity of discharge
[root](2gh). The discharge per second in this lamina is therefore
b[root](2gh) dh, and that of the whole jet is therefore
_
/h2
Q = | b [root](2gh) dh
_/h1

= 2/3 b[root](2g) {h2^(3/2) - h1^(3/2)}, (6)

where the first factor on the right is a coefficient depending on the
form of the orifice.

Now an orifice producing a rectangular jet must itself be very
approximately rectangular. Let B be the breadth, H1, H2, the depths to
the upper and lower edges of the orifice. Put

b [h2^(3/2) - h1^(3/2)] / B [H2^(3/2) - H1^(3/2)] = c. (7)

Then the discharge, in terms of the dimensions of the orifice, instead
of those of the jet, is

Q = (2/3)cB [root](2g) [H2^(3/2) - H1^(3/2)], (8)

the formula commonly given for the discharge of rectangular orifices.
The coefficient c is not, however, simply the coefficient of
contraction, the value of which is

b(h2 - h1)/B(H2 - H1),

and not that given in (7). It cannot be assumed, therefore, that c in
equation (8) is constant, and in fact it is found to vary for
different values of B/H2 and B/H1, and must be ascertained
experimentally.

_Relation between the Expressions (5) and (8)._--For a rectangular
orifice the area of the orifice is [omega] = B(H2 - H1), and the
depth measured to its centre is ½(H2 + H1). Putting these values in
(5),

Q1 = cB(H2 - H1) [root]{g(H2 + H1)}.

From (8) the discharge is

Q2 = (2/3)cB [root](2g) [H2^(3/2) - H1^(3/2)].

Hence, for the same value of c in the two cases,

Q2/Q1 = (2/3)[H2^(3/2) - H1^(3/2)] / [(H2 - H1)[root]{(H2 + H1)/2}].

Let H1/H2 = [sigma], then

Q2/Q1 = 0.9427(1 - [sigma]^(3/2)) /
{1 - [sigma] [root]{(1 + [sigma])}}. (9)

If H1 varies from 0 to [infinity], [sigma]( = H1/H2) varies from 0 to
1. The following table gives values of the two estimates of the
discharge for different values of [sigma]:--

+------------------+--------+------------------+--------+
| H1/H2 = [sigma]. | Q2/Q1. | H1/H2 = [sigma]. | Q2/Q1. |
+------------------+--------+------------------+--------+
| 0.0 | .943 | 0.8 | .999 |
| 0.2 | .979 | 0.9 | .999 |
| 0.5 | .995 | 1.0 | 1.000 |
| 0.7 | .998 | | |
+------------------+--------+------------------+--------+

Hence it is obvious that, except for very small values of [sigma], the
simpler equation (5) gives values sensibly identical with those of
(8). When [sigma]<0.5 it is better to use equation (8) with values of
c determined experimentally for the particular proportions of orifice
which are in question.

§ 40. _Large Jets having a Circular Section from Orifices in a
Vertical Plane Surface._--Let fig. 44 represent the section of the
jet, OO being the free surface level in the reservoir. The discharge
through the horizontal strip aabb, of breadth aa = b, between the
depths h1 + y and h1 + y + dy, is

dQ = b [root]{2g(h1 + y)} dy.

The whole discharge of the jet is
_
/d
Q = | b [root]{2g(h1 + y)} dy.
_/0

But b = d sin [phi]; y = ½d(1 - cos [phi]); dy = ½d sin [phi] d[phi].
Let [epsilon] = d/(2h1 + d), then
_
/[pi]
Q = ½d² [root]{2g(h1 + d/2)} | sin² [phi][root]{1 - [epsilon] cos [phi]} d[phi].
_/0

From eq. (5), putting [omega] = [pi]d²/4, h = h1 + d/2, c = 1 when d
is the diameter of the jet and not that of the orifice,

Q1 = ¼[pi]d² [root]{2g (h1 + d/2)},
_
/[pi]
Q/Q1 = 2/[pi] | sin² [phi] [root]{1 - [epsilon] cos [phi]} d[phi].
_/0

For

h1 = [infinity], [epsilon] = 0 and Q/Q1 = 1;

and for

h1 = 0, [epsilon] = 1 and Q/Q1 = 0.96.

So that in this case also the difference between the simple formula
(5) and the formula above, in which the variation of head at different
parts of the orifice is taken into account, is very small.

NOTCHES AND WEIRS

§ 41. _Notches, Weirs and Byewashes._--A notch is an orifice extending
up to the free surface level in the reservoir from which the discharge
takes place. A weir is a structure over which the water flows, the
discharge being in the same conditions as for a notch. The formula of
discharge for an orifice of this kind is ordinarily deduced by putting
H1 = 0 in the formula for the corresponding orifice, obtained as in
the preceding section. Thus for a rectangular notch, put H1 = 0 in
(8). Then

Q = (2/3)cB [root](2g) H^(3/2), (11)

where H is put for the depth to the crest of the weir or the bottom of
the notch. Fig. 45 shows the mode in which the discharge occurs in the
case of a rectangular notch or weir with a level crest. As, the free
surface level falls very sensibly near the notch, the head H should be
measured at some distance back from the notch, at a point where the
velocity of the water is very small.

Since the area of the notch opening is BH, the above formula is of the
form

Q = c × BH × k [root](2gH),

where k is a factor depending on the form of the notch and expressing
the ratio of the mean velocity of discharge to the velocity due to the
depth H.

§ 42. _Francis's Formula for Rectangular Notches._--The jet discharged
through a rectangular notch has a section smaller than BH, (a) because
of the fall of the water surface from the point where H is measured
towards the weir, (b) in consequence of the crest contraction, (c) in
consequence of the end contractions. It may be pointed out that while
the diminution of the section of the jet due to the surface fall and
to the crest contraction is proportional to the length of the weir,
the end contractions have nearly the same effect whether the weir is
wide or narrow.

J. B. Francis's experiments showed that a perfect end contraction,
when the heads varied from 3 to 24 in., and the length of the weir was
not less than three times the head, diminished the effective length of
the weir by an amount approximately equal to one-tenth of the head.
Hence, if l is the length of the notch or weir, and H the head
measured behind the weir where the water is nearly still, then the
width of the jet passing through the notch would be l - 0.2H, allowing
for two end contractions. In a weir divided by posts there may be more
than two end contractions. Hence, generally, the width of the jet is l
- 0.1nH, where n is the number of end contractions of the stream. The
contractions due to the fall of surface and to the crest contraction
are proportional to the width of the jet. Hence, if cH is the
thickness of the stream over the weir, measured at the contracted
section, the section of the jet will be c(l - 0.1nH)H and (§ 41) the
mean velocity will be 2/3 [root](2gH). Consequently the discharge
will be given by an equation of the form

Q = (2/3)c (l - 0.1nH)H [root](2gH)
= 5.35c (l - 0.1nH) H^(3/2).

This is Francis's formula, in which the coefficient of discharge c is
much more nearly constant for different values of l and h than in the
ordinary formula. Francis found for c the mean value 0.622, the weir
being sharp-edged.

§ 43. _Triangular Notch_ (fig. 46).--Consider a lamina issuing between
the depths h and h + dh. Its area, neglecting contraction, will be
bdh, and the velocity at that depth is [root](2gh). Hence the
discharge for this lamina is

b[root](2gh) dh.

But

B/b = H/(H - h); b = B(H - h)/H.

Hence discharge of lamina

= B(H - h) [root](2gh) dh/H;

and total discharge of notch
_
/H
= Q = B[root](2g) | (H - h)h^(½) dh/H
_/0

= (4/15) B[root](2g)H^(3/2).

or, introducing a coefficient to allow for contraction,

Q = (4/15)cB [root](2g) H^(½),

When a notch is used to gauge a stream of varying flow, the ratio B/H
varies if the notch is rectangular, but is constant if the notch is
triangular. This led Professor James Thomson to suspect that the
coefficient of discharge, c, would be much more constant with
different values of H in a triangular than in a rectangular notch, and
this has been experimentally shown to be the case. Hence a triangular
notch is more suitable for accurate gaugings than a rectangular notch.
For a sharp-edged triangular notch Professor J. Thomson found c =
0.617. It will be seen, as in § 41, that since ½BH is the area of
section of the stream through the notch, the formula is again of the
form

Q = c × ½BH × k[root](2gH),

where k = 8/15 is the ratio of the mean velocity in the notch to the
velocity at the depth H. It may easily be shown that for all notches
the discharge can be expressed in this form.

_Coefficients for the Discharge over Weirs, derived from the
Experiments of T. E. Blackwell. When more than one experiment was
made with the same head, and the results were pretty uniform, the
resulting coefficients are marked with an (*). The effect of the
converging wing-boards is very strongly marked._

+----------+-------------+---------------------------------+-----------------------------------------+
| | | Planks 2 in. thick, | |
| Heads in | Sharp Edge. | square on Crest. | Crests 3 ft. wide. |
| inches +------+------+-----+-----+-------+-------------+------+------+------+------+------+------+
| measured | | | | | |10 ft. long, | 3 ft.| 3 ft.| 3 ft.| 6 ft.|10 ft.|10 ft.|
|from still| 3 ft.|10 ft.|3 ft.|6 ft.| 10 ft.| wing-boards | long,| long,| long,| long,| long,| long,|
| Water in | long.| long.|long.|long.| long. | making an |level.|fall 1|fall 1|level.|level.|fall 1|
|Reservoir.| | | | | |angle of 60°.| |in 18.|in 12.| | |in 18.|
+----------+------+------+-----+-----+-------+-------------+------+------+------+------+------+------+
| 1 | .677 | .809 |.467 |.459 |.435[4]| .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 | .617 | .656 |.549 |.575 |.602* | .656 | .419 | .431 | .455 | .497*| .. | .515 |
| 5 | .602 | .650*|.588 |.601*|.609* | .671 | .479 | .516 | .. | .. | .518 | .. |
| 6 | .593 | .. |.593*|.608*|.576* | .. | .501*| .. | .531 | .507 | .513 | .543 |
| 7 | .. | .. |.617*|.608*|.576* | .. | .488 | .513 | .527 | .497 | .. | .. |
| 8 | .. | .581 |.606*|.590*|.548* | .. | .470 | .491 | .. | .. | .468 | .507 |
| 9 | .. | .530 |.600 |.569*|.558* | .. | .476 | .492*| .498 | .480*| .486 | .. |
| 10 | .. | .. |.614*|.539 |.534* | .. | .. | .. | .. | .465*| .455 | .. |
| 12 | .. | .. | .. |.525 |.534* | .. | .. | .. | .. | .467*| .. | .. |
| 14 | .. | .. | .. |.549*| .. | .. | .. | .. | .. | .. | .. | .. |
+----------+------+------+-----+-----+-------+-------------+------+------+------+------+------+------+

§ 44. _Weir with a Broad Sloping Crest._--Suppose a weir formed with a
broad crest so sloped that the streams flowing over it have a movement
sensibly rectilinear and uniform (fig. 47). Let the inner edge be so
rounded as to prevent a crest contraction. Consider a filament aa´,
the point a being so far back from the weir that the velocity of
approach is negligible. Let OO be the surface level in the reservoir,
and let a be at a height h´´ below OO, and h´ above a´. Let h be the
distance from OO to the weir crest and e the thickness of the stream
upon it. Neglecting atmospheric pressure, which has no influence, the
pressure at a is Gh´´; at a´ it is Gz. If v be the velocity at a´,

v²/2g = h´ + h´´ - z = h - e;

Q = be [root]{2g(h - e)}.

Theory does not furnish a value for e, but Q = 0 for e = 0 and for e =
h. Q has therefore a maximum for a value of e between 0 and h,
obtained by equating dQ/de to zero. This gives e = (2/3)h, and,
inserting this value,

Q = 0.385 bh [root](2gh),

as a maximum value of the discharge with the conditions assigned.
Experiment shows that the actual discharge is very approximately equal
to this maximum, and the formula is more legitimately applicable to
the discharge over broad-crested weirs and to cases such as the
discharge with free upper surface through large masonry sluice
openings than the ordinary weir formula for sharp-edged weirs. It
should be remembered, however, that the friction on the sides and
crest of the weir has been neglected, and that this tends to reduce a
little the discharge. The formula is equivalent to the ordinary weir
formula with c = 0.577.

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 h1, 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,

[h] = u²/2g

has been somewhere expended in producing the velocity u, and hence the
true heads measured in still water would have been h1 + [h] and h2 +
[h]. Consequently the discharge, allowing for the velocity of
approach, is

Q = (2/3)cb [root](2g) {(h2 + [h])^(3/2) - (h1 + [h])^(3/2)}. (1)

And for a rectangular notch for which h1 = 0, the discharge is

Q = (2/3)cb [root](2g) {(h2 + [h])^(3/2) - [h]^(3/2)}. (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 [Omega] be the sectional area of the channel where h1
and h2 are measured. Then u = Q/[Omega] and [h] = Q²/2g [Omega]².

This value introduced in the equations above would render them
excessively cumbrous. In cases therefore where [Omega] only is known,
it is best to proceed by approximation. Calculate an approximate value
Q´ of Q by the equation

Q´ = (2/3)cb [root](2g) {h2^(3/2) - h1^(3/2)}.

Then [h] = Q´²/2g[Omega]² nearly. This value of [h] 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 M1m1 (fig. 49) in the upper part of the orifice
issues with a head h´ which may have any value between h1 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 Q1, Q2 are the discharges
from the upper and lower parts of the orifice, b the width of the
orifice, then

Q1 = (2/3)cb [root](2g) {h^(3/2) - h1^(3/2)}
(3)
Q1 = cb (h2 - h) [root](2gh).

In the case of a rectangular notch or weir, h1 = 0. Inserting this
value, and adding the two portions of the discharge together, we get
for a drowned weir

Q = cb[root](2gh) (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. 50).

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

0.1 0.629 0.7 0.578
0.2 0.614 0.8 0.583
0.3 0.600 0.9 0.596
0.4 0.590 0.95 0.607
0.5 0.582 1.00 0.628
0.6 0.578

If velocity of approach is taken into account, let [h] be the
head due to that velocity; then, adding [h] to each of the
heads in the equations (3), and reducing, we get for a weir

Q = cb [root]{2g} [(h2 + [h]) (h + [h])^(½) - (1/3)(h + [h])^(3/2)
- (2/3)[h]^(3/2)]; (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 flow
of the stream, which causes a difference of surface-level above and
below the pier (fig. 51). If it is necessary to estimate this
difference of level, the flow 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.

§ 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
Chaussées_ (October 1888, January 1890, 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 × 1.0 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 0.6 metres. From this extends a channel 200 metres in length with a
slope of 1 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 Weir._--The weir crest was 3.72 ft. above the bottom of the
canal and formed by a plate ¼ in. thick. It was sharp-edged 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 = mlh [root](2gh), (1)

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 |
| 0.10 | .328 | 0.4336 |
| 0.15 | .492 | 0.4284 |
| 0.20 | .656 | 0.4262 |
| 0.25 | .820 | 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 | 1.640 | 0.4313 |
| 0.55 | 1.804 | 0.4327 |
| 0.60 | 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 Approach._--To take account of the velocity
of approach u it is usual to replace h in the formula by h + au²/2g
where [alpha] is a coefficient not very well ascertained. Then

Q = [mu]l (h + [alpha]u²/2g) [root]{2g(h + [alpha]u²/2g)}
= [mu]lh [root](2gh)(1 + [alpha]u²/2gh)^(3/2). (2)

The original simple equation can be used if

m = [mu](1 + [alpha]u²/2gh)^(3/2)

or very approximately, since u²/2gh is small,

m = [mu](1 + (3/2)[alpha]u²/2gh). (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 (1)

u²/2gh = Q²/{2ghl²(p + h)²} = m²{h/(p + h)}², (4)

so that (3) may be written

m = [mu][1 + k{h/(p + h)}²]. (5)

Gaugings were made with weirs of 0.75, 0.50, 0.35, and 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
(1):--

m = [mu](1 + 2.5u²/2gh)
= [mu][1 + 0.55 {h/(p + h)}²].

Values of [mu]--

+----------------+--------------+--------+
| Head h metres. | Head h feet. | [mu] |
+----------------+--------------+--------+
| 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 | 1.640 | 0.4118 |
| 0.60 | 1.968 | 0.4092 |
+----------------+--------------+--------+

An approximate formula for [mu] is:

[mu] = 0.405 + 0.003/h (h in metres)

[mu] = 0.405 + 0.01/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.

Coefficient.
Inclination up stream 1 to 1 0.93
" " 3 to 2 0.94
" " 3 to 1 0.96
Vertical weir 1.00
Inclination down stream 3 to 1 1.04
" " 3 to 2 1.07
" " 1 to 1 1.10
" " 1 to 2 1.12
" " 1 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 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 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 clear from the
upstream edge or may adhere to the flat crest falling free beyond the
down-stream edge. In the former case the condition is that of a
sharp-edged 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 1½ 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 = m1 [0.70 + 0.185h/c], where m1 is the coefficient for a
sharp-edged weir in similar conditions. Rounding the upstream edge
even to a small extent modifies the discharge. If R is the radius of
the rounding the coefficient m is increased in the ratio 1 to 1 + R/h
nearly. The results are limited to R less than ½ in.

_Drowned Weirs._--Let h (fig. 54) be the height of head water and h1
that of tail water above the weir crest. Then Bazin obtains as the
approximate formula for the coefficient of discharge

m = 1.05m1 [1 + (1/5)h1/p] [root 3]{(h - h1)/h},

where as before m1 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-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 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 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

(2/3)[root](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 = ½gt², x = (2/3)[root](2gh) t;

.: y = (9/16)x²/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 and equal to ½g on any scale; af horizontally and equal to
2/3 [root](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 l½ 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 one-third 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 [Omega] be the area of the mouthpiece, [omega] the area
of the stream at the contracted section EF. Let v, p be the velocity
and pressure at EF, and v1, p1 the same quantities at GH. If the
discharge is into the air, p1 is equal to the atmospheric pressure
p_a.

The total head of any filament which goes to form the jet, taken at a
point where its velocity is sensibly zero, is h + p_a/G; at EF the
total head is v²/2g + p/G; at GH it is v1²/2g + p1/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 - v1)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_a/G = v²/2g + p/G = v1²/2g + p1/G + (v - v1)²/2g.

But [omega]v = [Omega]v1, and [omega] = c_c[Omega], if c_c is the
coefficient of contraction within the mouthpiece. Hence

v = [Omega]v1/[omega] = v1/c_c.

Supposing the discharge into the air, so that p1 = p_a,

h + p_a/G = v1²/2g + p_a/G + (v1²/2g)(1/c_c - 1)²;

(v1/2g){1 + (1/c_c - 1)²} = h;

.: v1 = [root](2gh)/[root]{1 + (1/c_c - 1)²}; (1)

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_c = 0.64, the value for simple orifices, then
the coefficient of velocity is

c_v = 1/[root]{1 + (1/c_c - 1)²} = 0.87 (2)

The actual value of c_v, found by experiment is 0.82, which does not
differ more from the theoretical value than might be expected if the
friction of the mouthpiece is allowed for. Hence, for mouthpieces of
this kind, and for the section at GH,

c_v = 0.82 c_c = 1.00 c = 0.82,

Q = 0.82[Omega] [root](2gh).

It is easy to see from the equations that the pressure p at EF is less
than atmospheric pressure. Eliminating v1, we get

(p_a - p)/G = ¾h nearly; (3)

or

p = p_a - ¾Gh lb. 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_a - p)/G = ¾h 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.

§ 50. _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 = c_v c_c[Omega] [root](2gh), (4)

where [Omega] is the area of the external end of the mouthpiece, and
c_c[Omega] the section of the contracted jet beyond the mouthpiece.

_Convergent Mouthpieces (Castel's Experiments).--Smallest diameter of
orifice = 0.05085 ft. Length of mouthpiece = 2.6 Diameters._

+------------+--------------+--------------+--------------+
| |Coefficient of|Coefficient of|Coefficient of|
| Angle of | Contraction, | Velocity, | Discharge, |
|Convergence.| c_c | c_v | c |
+------------+--------------+--------------+--------------+
| 0° 0´ | .999 | .830 | .829 |
| 1° 36´ | 1.000 | .866 | .866 |
| 3° 10´ | 1.001 | .894 | .895 |
| 4° 10´ | 1.002 | .910 | .912 |
| 5° 26´ | 1.004 | .920 | .924 |
| 7° 52´ | .998 | .931 | .929 |
| 8° 58´ | .992 | .942 | .934 |
| 10° 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° 0´ | .945 | .971 | .918 |
| 23° 0´ | .937 | .974 | .913 |
| 29° 58´ | .919 | .975 | .896 |
| 40° 20´ | .887 | .980 | .869 |
| 48° 50´ | .861 | .984 | .847 |
+------------+--------------+--------------+--------------+

The maximum coefficient of discharge is that for a mouthpiece with a
convergence of 13°24´.

The values of c_v and c_c 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, c_c diminishes, and on the other hand the loss
of energy diminishes, so that c_v increases with the convergence,
there is an angle for which the product c_c c_v, 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. 60. Suppose that at EF it
becomes cylindrical, so that the jet may be taken to be of the
diameter EF. Let [omega], v, p be the section, velocity and pressure
at CD, and [Omega], v1, p1 the same quantities at EF, p_a 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 + p_a/G = v²/2g + p/G = v1²/2g + p1/G.

If the jet discharges into the air, p1 = p_a; and

v1²/2g = h;

v1 = [root](2gh);

or, if a coefficient is introduced to allow for friction,

v1 = c_v [root](2gh);

where c_v is about 0.97 if the mouthpiece is smooth and well formed.

Q = [Omega] v1 = c_v [Omega] [root](2gh).

Hence the discharge depends on the area of the stream at EF, and not
at 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 = 0, the continuity of flow is impossible. In fact the stream
disengages itself from the mouthpiece for some value of p greater than
0 (fig. 61).

From the equations,

p/G = p_a/G = (v² - v1²)/2g.

Let [Omega]/[omega] = m. Then

v = v1m;

p/G = p_a/G - v1²(m² - 1)/2g
= p_a/G - (m² - 1)h;

whence we find that p/G will become zero or negative if

[Omega]/[omega] >= [root]{(h + p_a/G)/h}
= [root]{1 + p_a/Gh};

or, putting p_a/G = 34 ft., if

[Omega]/[omega] >= [root]{(h + 34)/h}.

In practice there will be an interruption of the full bore flow with a
less ratio of [Omega]/[omega], 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 = [omega] [root]{2g(h + p_a/G)};

that is, the discharge is the same as for a well-bell-mouthed
mouthpiece of area [omega], 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
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 by the discharge pipe
C.

Fig. 63 shows the whole arrangement in a diagrammatic way. A is the
reservoir which supplies the water that effects the pumping; B is the
reservoir of water to be pumped; C is the reservoir into which the
water is pumped.

DISCHARGE WITH VARYING HEAD

§ 53. _Flow from a Vessel when the Effective Head varies with the
Time._--Various useful problems arise relating to the time of emptying
and filling vessels, reservoirs, lock chambers, &c., where the flow is
dependent on a head which increases or diminishes during the
operation. The simplest of these problems is the case of filling or
emptying a vessel of constant horizontal section.

_Time of Emptying or Filling a Vertical-sided Lock Chamber._--Suppose
the lock chamber, which has a water surface of [Omega] square ft., is
emptied through a sluice in the tail gates, of area [omega], placed
below the tail-water level. Then the effective head producing flow
through the sluice is the difference of level in the chamber and tail
bay. Let H (fig. 64) be the initial difference of level, h the
difference of level after t seconds. Let -dh be the fall of level in
the chamber during an interval dt. Then in the time dt the volume in
the chamber is altered by the amount -[Omega]dh, and the outflow from
the sluice in the same time is c[omega][root](2gh)dt. Hence the
differential equation connecting h and t is

c[omega] [root](2gh) dt + [Omega]h = 0.

For the time t, during which the initial head H diminishes to any
other value h,
_ _
/h /t
-{[Omega]/(c[omega] [root]2g)} | dh/[root]h = | dt.
_/H _/0

.: t = 2[Omega]([root]H - [root]h) / {c[omega] [root](2g)}
= ([Omega]/c[omega]){[root](2H/g) - [root](2h/g)}.

For the whole time of emptying, during which h diminishes from H to 0,

T = ([Omega]/c[omega]) [root](2H/g).

Comparing this with the equation for flow under a constant head, it
will be seen that the time is double that required for the discharge
of an equal volume under a constant head.

The time of filling the lock through a sluice in the head gates is
exactly the same, if the sluice is below the tail-water level. But if
the sluice is above the tail-water level, then the head is constant
till the level of the sluice is reached, and afterwards it diminishes
with the time.

PRACTICAL USE OF ORIFICES IN GAUGING WATER

§ 54. If the water to be measured is passed through a known orifice
under an arrangement by which the constancy of the head is ensured,
the amount which passes in a given time can be ascertained by the
formulae already given. It will obviously be best to make the orifices
of the forms for which the coefficients are most accurately
determined; hence sharp-edged orifices or notches are most commonly
used.

_Water Inch._--For measuring small quantities of water circular
sharp-edged orifices have been used. The discharge from a circular
orifice one French inch in diameter, with a head of one line above the
top edge, was termed by the older hydraulic writers a water-inch. A
common estimate of its value was 14 pints per minute, or 677 English
cub. ft. in 24 hours. An experiment by C. Bossut gave 634 cub. ft. in
24 hours (see Navier's edition of _Belidor's Arch. Hydr._, p. 212).