where c = 0.62; or, better, the formula in § 42 may be used.
Gauging weirs are most commonly in the form of rectangular notches;
and care should be taken that the crest is accurately horizontal, and
that the weir is normal to the direction of flow of the stream. If the
planks are thick, they should be bevelled (fig. 67), and then the edge
may be protected by a metal plate about (1/10)th in. thick to secure
the requisite accuracy of form and sharpness of edge. In permanent
gauging weirs, a cast steel plate is sometimes used to form the edge
of the weir crest. The weir should be large enough to discharge the
maximum volume flowing in the stream, and at the same time it is
desirable that the minimum head should not be too small (say half a
foot) to decrease the effects of errors of measurement. The section of
the jet over the weir should not exceed one-fifth the section of the
stream behind the weir, or the velocity of approach will need to be
taken into account. A triangular notch is very suitable for
measurements of this kind.
[Illustration: FIG. 66.]
If the flow is variable, the head h must be recorded at equidistant
intervals of time, say twice daily, and then for each 12-hour period
the discharge must be calculated for the mean of the heads at the
beginning and end of the time. As this involves a good deal of
troublesome calculation, E. Sang proposed to use a scale so graduated
as to read off the discharge in cubic feet per second. The lengths of
the principal graduations of such a scale are easily calculated by
putting Q = 1, 2, 3 ... in the ordinary formulae for notches; the
intermediate graduations may be taken accurately enough by subdividing
equally the distances between the principal graduations.
[Illustration: FIG. 67.]
[Illustration: FIG. 68.]
The accurate measurement of the discharge of a stream by means of a
weir is, however, in practice, rather more difficult than might be
inferred from the simplicity of the principle of the operation. Apart
from the difficulty of selecting a suitable coefficient of discharge,
which need not be serious if the form of the weir and the nature of
its crest are properly attended to, other difficulties of measurement
arise. The length of the weir should be very accurately determined,
and if the weir is rectangular its deviations from exactness of level
should be tested. Then the agitation of the water, the ripple on its
surface, and the adhesion of the water to the scale on which the head
is measured, are liable to introduce errors. Upon a weir 10 ft. long,
with 1 ft. depth of water flowing over, an error of 1-1000th of a foot
in measuring the head, or an error of 1-100th of a foot in measuring
the length of the weir, would cause an error in computing the
discharge of 2 cub. ft. per minute.
_Hook Gauge._--For the determination of the surface level of water,
the most accurate instrument is the hook gauge used first by U. Boyden
of Boston, in 1840. It consists of a fixed frame with scale and
vernier. In the instrument in fig. 68 the vernier is fixed to the
frame, and the scale slides vertically. The scale carries at its lower
end a hook with a fine point, and the scale can be raised or lowered
by a fine pitched screw. If the hook is depressed below the water
surface and then raised by the screw, the moment of its reaching the
water surface will be very distinctly marked, by the reflection from a
small capillary elevation of the water surface over the point of the
hook. In ordinary light, differences of level of the water of .001 of
a foot are easily detected by the hook gauge. If such a gauge is used
to determine the heads at a weir, the hook should first be set
accurately level with the weir crest, and a reading taken. Then the
difference of the reading at the water surface and that for the weir
crest will be the head at the weir.
§ 56. _Modules used in Irrigation._--In distributing water for
irrigation, the charge for the water may be simply assessed on the
area of the land irrigated for each consumer, a method followed in
India; or a regulated quantity of water may be given to each consumer,
and the charge may be made proportional to the quantity of water
supplied, a method employed for a long time in Italy and other parts
of Europe. To deliver a regulated quantity of water from the
irrigation channel, arrangements termed modules are used. These are
constructions intended to maintain a constant or approximately
constant head above an orifice of fixed size, or to regulate the size
of the orifice so as to give a constant discharge, notwithstanding the
variation of level in the irrigating channel.
[Illustration: FIG. 69.]
§ 57. _Italian Module._--The Italian modules are masonry
constructions, consisting of a regulating chamber, to which water is
admitted by an adjustable sluice from the canal. At the other end of
the chamber is an orifice in a thin flagstone of fixed size. By means
of the adjustable sluice a tolerably constant head above the fixed
orifice is maintained, and therefore there is a nearly constant
discharge of ascertainable amount through the orifice, into the
channel leading to the fields which are to be irrigated.
[Illustration: FIG. 70.--Scale 1/100.]
In fig. 69, A is the adjustable sluice by which water is admitted to
the regulating chamber, B is the fixed orifice through which the water
is discharged. The sluice A is adjusted from time to time by the canal
officers, so as to bring the level of the water in the regulating
chamber to a fixed level marked on the wall of the chamber. When
adjusted it is locked. Let [omega]1 be the area of the orifice through
the sluice at A, and [omega]2 that of the fixed orifice at B; let h1
be the difference of level between the surface of the water in the
canal and regulating chamber; h2 the head above the centre of the
discharging orifice, when the sluice has been adjusted and the flow
has become steady; Q the normal discharge in cubic feet per second.
Then, since the flow through the orifices at A and B is the same,
Q = c1[omega]1 [root](2gh1) = c2[omega]2 [root](2gh2),
where c1 and c2 are the coefficients of discharge suitable for the two
orifices. Hence
c1[omega]1/c2[omega]2 = [root](h2/h1).
If the orifice at B opened directly into the canal without any
intermediate regulating chamber, the discharge would increase for a
given change of level in the canal in exactly the same ratio.
Consequently the Italian module in no way moderates the fluctuations
of discharge, except so far as it affords means of easy adjustment
from time to time. It has further the advantage that the cultivator,
by observing the level of the water in the chamber, can always see
whether or not he is receiving the proper quantity of water.
On each canal the orifices are of the same height, and intended to
work with the same normal head, the width of the orifices being varied
to suit the demand for water. The unit of discharge varies on
different canals, being fixed in each case by legal arrangements. Thus
on the Canal Lodi the unit of discharge or one module of water is the
discharge through an orifice 1.12 ft. high, 0.12416 ft. wide, with a
head of 0.32 ft. above the top edge of the orifice, or .88 ft. above
the centre. This corresponds to a discharge of about 0.6165 cub. ft.
per second.
[Illustration: FIG. 71.]
In the most elaborate Italian modules the regulating chamber is arched
over, and its dimensions are very exactly prescribed. Thus in the
modules of the Naviglio Grande of Milan, shown in fig. 70, the
measuring orifice is cut in a thin stone slab, and so placed that the
discharge is into the air with free contraction on all sides. The
adjusting sluice is placed with its sill flush with the bottom of the
canal, and is provided with a rack and lever and locking arrangement.
The covered regulating chamber is about 20 ft. long, with a breadth
1.64 ft. greater than that of the discharging orifice. At precisely
the normal level of the water in the regulating chamber, there is a
ceiling of planks intended to still the agitation of the water. A
block of stone serves to indicate the normal level of the water in the
chamber. The water is discharged into an open channel 0.655 ft. wider
than the orifice, splaying out till it is 1.637 ft. wider than the
orifice, and about 18 ft. in length.
§ 58. _Spanish Module._--On the canal of Isabella II., which supplies
water to Madrid, a module much more perfect in principle than the
Italian module is employed. Part of the water is supplied for
irrigation, and as it is very valuable its strict measurement is
essential. The module (fig. 72) consists of two chambers one above the
other, the upper chamber being in free communication with the
irrigation canal, and the lower chamber discharging by a culvert to
the fields. In the arched roof between the chambers there is a
circular sharp-edged orifice in a bronze plate. Hanging in this there
is a bronze plug of variable diameter suspended from a hollow brass
float. If the water level in the canal lowers, the plug descends and
gives an enlarged opening, and conversely. Thus a perfectly constant
discharge with a varying head can be obtained, provided no clogging or
silting of the chambers prevents the free discharge of the water or
the rise and fall of the float. The theory of the module is very
simple. Let R (fig. 71) be the radius of the fixed opening, r the
radius of the plug at a distance h from the plane of flotation of the
float, and Q the required discharge of the module. Then
Q = c[pi](R² - r²) [root](2gh).
Taking c = 0.63,
Q = 15.88(R² - r²) [root]h;
r = [root]{R² - Q/15.88 [root]h}.
Choosing a value for R, successive values of r can be found for
different values of h, and from these the curve of the plug can be
drawn. The module shown in fig. 72 will discharge 1 cubic metre per
second. The fixed opening is 0.2 metre diameter, and the greatest head
above the fixed orifice is 1 metre. The use of this module involves a
great sacrifice of level between the canal and the fields. The module
is described in Sir C. Scott-Moncrieff's _Irrigation in Southern
Europe_.
§ 59. _Reservoir Gauging Basins._--In obtaining the power to store the
water of streams in reservoirs, it is usual to concede to riparian
owners below the reservoirs a right to a regulated supply throughout
the year. This compensation water requires to be measured in such a
way that the millowners and others interested in the matter can assure
themselves that they are receiving a proper quantity, and they are
generally allowed a certain amount of control as to the times during
which the daily supply is discharged into the stream.
[Illustration: FIG. 72.]
Fig. 74 shows an arrangement designed for the Manchester water works.
The water enters from the reservoir at chamber A, the object of which
is to still the irregular motion of the water. The admission is
regulated by sluices at b, b, b. The water is discharged by orifices
or notches at a, a, over which a tolerably constant head is maintained
by adjusting the sluices at b, b, b. At any time the millowners can
see whether the discharge is given and whether the proper head is
maintained over the orifices. To test at any time the discharge of the
orifices, a gauging basin B is provided. The water ordinarily flows
over this, without entering it, on a floor of cast-iron plates. If the
discharge is to be tested, the water is turned for a definite time
into the gauging basin, by suddenly opening and closing a sluice at c.
The volume of flow can be ascertained from the depth in the gauging
chamber. A mechanical arrangement (fig. 73) was designed for securing
an absolutely constant head over the orifices at a, a. The orifices
were formed in a cast-iron plate capable of sliding up and down,
without sensible leakage, on the face of the wall of the chamber. The
orifice plate was attached by a link to a lever, one end of which
rested on the wall and the other on floats f in the chamber A. The
floats rose and fell with the changes of level in the chamber, and
raised and lowered the orifice plate at the same time. This mechanical
arrangement was not finally adopted, careful watching of the sluices
at b, b, b, being sufficient to secure a regular discharge. The
arrangement is then equivalent to an Italian module, but on a large
scale.
[Illustration: FIG. 73.--Scale 1/120.]
[Illustration: FIG. 74.--Scale 1/500.]
§ 60. _Professor Fleeming Jenkin's Constant Flow Valve._--In the
modules thus far described constant discharge is obtained by varying
the area of the orifice through which the water flows. Professor F.
Jenkin has contrived a valve in which a constant pressure head is
obtained, so that the orifice need not be varied (_Roy. Scot. Society_
_of Arts_, 1876). Fig. 75 shows a valve of this kind suitable for a
6-in. water main. The water arriving by the main C passes through an
equilibrium valve D into the chamber A, and thence through a sluice O,
which can be set for any required area of opening, into the
discharging main B. The object of the arrangement is to secure a
constant difference of pressure between the chambers A and B, so that
a constant discharge flows through the stop valve O. The equilibrium
valve D is rigidly connected with a plunger P loosely fitted in a
diaphragm, separating A from a chamber B2 connected by a pipe B1 with
the discharging main B. Any increase of the difference of pressure in
A and B will drive the plunger up and close the equilibrium valve, and
conversely a decrease of the difference of pressure will cause the
descent of the plunger and open the equilibrium valve wider. Thus a
constant difference of pressure is obtained in the chambers A and B.
Let [omega] be the area of the plunger in square feet, p the
difference of pressure in the chambers A and B in pounds per square
foot, w the weight of the plunger and valve. Then if at any moment
p[omega] exceeds w the plunger will rise, and if it is less than w the
plunger will descend. Apart from friction, and assuming the valve D to
be strictly an equilibrium valve, since [omega] and w are constant, p
must be constant also, and equal to w/[omega]. By making w small and
[omega] large, the difference of pressure required to ensure the
working of the apparatus may be made very small. Valves working with a
difference of pressure of ½ in. of water have been constructed.
[Illustration: FIG. 75.--Scale 1/24.]