[Illustration: FIG. 32.]
Hence by a simple observation of pressure difference, the flow in the
main at any moment can be determined. Notice that the pressure height
at C will be the same as at A except for a small loss h_f due to
friction and eddying between A and B. To get the pressure at the
throat very exactly Herschel surrounds it by an annular passage
communicating with the throat by several small holes, sometimes formed
in vulcanite to prevent corrosion. Though constructed to prevent
eddying as much as possible there is some eddy loss. The main effect
of this is to cause a loss of head between A and C which may vary from
a fraction of a foot to perhaps 5 ft. at the highest velocities at
which a meter can be used. The eddying also affects a little the
Venturi head h. Consequently an experimental coefficient must be
determined for each meter by tank measurement. The range of this
coefficient is, however, surprisingly small. If to allow for friction,
u = k[root]{[rho]²/([rho]² - 1)}[root](2gh), then Herschel found
values of k from 0.97 to 1.0 for throat velocities varying from 8 to
28 ft. per sec. The meter is extremely convenient. At Staines
reservoirs there are two meters of this type on mains 94 in. in
diameter. Herschel contrived a recording arrangement which records the
variation of flow from hour to hour and also the total flow in any
given time. In Great Britain the meter is constructed by G. Kent, who
has made improvements in the recording arrangement.
[Illustration: FIG. 33.]
In the Deacon Waste Water Meter (fig. 33) a different principle is
used. A disk D, partly counter-balanced by a weight, is suspended in
the water flowing through the main in a conical chamber. The
unbalanced weight of the disk is supported by the impact of the water.
If the discharge of the main increases the disk rises, but as it rises
its position in the chamber is such that in consequence of the larger
area the velocity is less. It finds, therefore, a new position of
equilibrium. A pencil P records on a drum moved by clockwork the
position of the disk, and from this the variation of flow is inferred.
§ 33. _Pressure, Velocity and Energy in Different Stream Lines._--The
equation of Bernoulli gives the variation of pressure and velocity
from point to point along a stream line, and shows that the total
energy of the flow across any two sections is the same. Two other
directions may be defined, one normal to the stream line and in the
plane containing its radius of curvature at any point, the other
normal to the stream line and the radius of curvature. For the
problems most practically useful it will be sufficient to consider the
stream lines as parallel to a vertical or horizontal plane. If the
motion is in a vertical plane, the action of gravity must be taken
into the reckoning; if the motion is in a horizontal plane, the terms
expressing variation of elevation of the filament will disappear.[3]
[Illustration: FIG. 34.]
Let AB, CD (fig. 34) be two consecutive stream lines, at present
assumed to be in a vertical plane, and PQ a normal to these lines
making an angle [phi] with the vertical. Let P, Q be two particles
moving along these lines at a distance PQ = ds, and let z be the
height of Q above the horizontal plane with reference to which the
energy is measured, v its velocity, and p its pressure. Then, if H is
the total energy at Q per unit of weight of fluid,
H = z + p/G + v²/2g.