Under this head three general cases are to be considered: 1, that of
liquids _issuing from orifices_; 2, _their flow through tubes_, or in
streams; and, 3, _the effects of the momentum and impact of liquids_.
The principles governing the action of the two last will be introduced,
2, under that portion of the work relating to “piping,” and, 3, under
the sections pertaining to the jet pump. Throughout the different
portions of the book will the three cases be still further elucidated.
Now, as to the laws governing the escape of liquids under pressure
through an opening, it may be understood that when the liquid escapes
from a vessel, owing to the excess of the internal pressure, _the
volume which escapes depends on the section of the orifice and the
velocity with which the liquid molecules move at the moment of their
escape from it_.
This velocity depends upon the density of the liquid, the excess of
pressure at the opening, and the friction of the liquid, both at the
opening and against the walls. When the aperture is made in a very
thin wall of a large vessel, so as to remove, as much as possible, the
causes tending to modify the motion of the escaping fluid, the laws
of the escape are comprised in the following theorem, discovered by
Torricelli, in 1643, as a consequence of the law of the fall of bodies
discovered by Galileo: “Liquid molecules, flowing from an orifice,
_have the same velocity, as if they fell freely in vacuo from a height
equal to the vertical distance from the surface to the center of the
orifice_.”
Deductions from the above:—1, _the velocity depends on the depth of
the orifice from the surface, and is independent of the density of the
liquid_. Water and mercury in vacuo would fall from the same height in
the same time; and so escaping from an orifice at the same depth, below
the surface, would pass out with equal velocity; but mercury, being
13·5 times as heavy as water, the pressure exerted at the aperture of a
vessel filled with mercury, will be 13·5 times as great as the pressure
exerted at the aperture of a vessel filled with water; 2, _the velocity
of liquids is as the square roots of the depths of the orifices below
the surfaces of the liquids_.
NOTE.—Torricelli discovered, in the early part of the 17th century,
the remarkable fact that a fluid issues from a small orifice with the
same velocity (friction and atmospheric resistance excluded) which it
would have acquired in falling through the depth from its surface.
This was one of a long series of discoveries leading toward the now
almost exact science of hydro-mechanics.
Thus stating, the velocity of a liquid escaping from an orifice one
foot below the surface to be _one_; from a similar orifice four feet
below the surface, it will be _two_, and at nine feet _three_, at
sixteen feet _four_, and so on.
From comparative experiments made by a great number of observers,
it is learned that the actual flow is only about two-thirds of the
theoretical flow. See note.
The form and constitution of liquid veins have been studied and found
to be:
1.—That the fluid issuing vertically from an orifice made in a plane
and thin horizontal wall, is always composed of two distinct parts,
Fig. 97, the portion nearest the orifice is calm and transparent, like
a rod of glass, gradually decreasing in diameter. The lower part, on
the contrary, is always agitated, and takes an irregular form, in which
are regularly distributed elongated swellings, called _ventres_, whose
maximum diameter is greater than that of the orifice.
NOTE.—_Theoretical and actual flow._—_The actual flow from an
orifice_, is the volume of liquid which escapes from it in a given
time. _The theoretical flow_, is a volume equal to that of a cylinder
which has for its base the orifice, and for its height the velocity,
furnished by the discovery of Torricelli. That is, the theoretical
flow is the product of the area of the orifice multiplied by the
theoretical velocity. It is observed that the vein escaping from an
orifice, contracts quite rapidly, so that its diameter is soon only
about two-thirds of the diameter of the orifice. If there was no
contraction of the vein after leaving the orifice, and its velocity
was the theoretical velocity, the actual flow would be the same
as that indicated by theory. But its section is much less than at
the orifice, and its velocity is not so great as the theoretical
velocity, so that the actual flow is much less than the theoretical
flow; and in order to reduce this to the first, it is necessary to
multiply it by a fraction.
2.—In the lower part of the vein, the liquid is not continuous; for if
we employ an opaque liquid, as mercury, we can see through the vein,
Fig. 98. The apparent continuity in a vein of water is owing to the
fact that the globules which constitute it succeed each other at a
distance inappreciable to the eye.
[Illustration: FIGS. 97, 98.]
_The time it takes by a vessel to empty itself_ is to the time
required, when it is kept constantly full, to discharge the same
quantity of water, as 2 to 1, and the spaces described by the surface
in its descent in a column of equal size throughout, are as the odd
numbers, 9, 7, 5, 3, 1. Thus these spaces measure equal times. Since
liquids are not perfectly mobile, and their exit at an orifice must be
retarded by cohesion and friction, the results thus far given are much
modified in practice.
When a liquid flows through an orifice in a vessel, eddies are formed
about the sides of the orifice, preventing the escape of a jet
equivalent to its full size; and owing to these, and to acceleration
of velocity, _if the jet be downward_, it rapidly contracts in its
diameter. At a distance outside about equal to diameter of the opening,
it is contracted to 2/3 or 5/7 its original area; and this part has
been called the “contracted vein.” It has been shown, that below this
the stream still contracts, though less rapidly.
These swellings separate more widely as they descend with increased
rapidity; but falling through great heights, the whole may finally be
dissipated in a mist.
NOTE.—The annular swellings contain air and arise from a periodical
succession of pulsations near the orifice, which must be produced by
very small oscillations of the entire mass of the liquid, so that the
velocity of the flow is periodically variable. The sucking, whistling
noise which is often heard in the descent of water through an orifice
is caused by air drawn in by the whirling motion. See Fig. 103.
If an orifice in a vessel looks downward, and the column of liquid over
it be short, this will simply drop out by its own weight, starting
at a velocity of o. But if a considerable depth of liquid be above,
its gravity produces a corresponding pressure on its base, or on that
liquid which is near it; so that, if a plug be removed from an orifice
in or close to the base, the liquid starts at once into rapid motion.
[Illustration: FIG. 99.]
Each particle of a jet A issuing from the side of a vessel moves
horizontally with the velocity above mentioned, _but it is at once
drawn downward by the force of gravity_ in the same manner as a bullet
fired from a gun, with its axis horizontal. It is well known that the
bullet describes _a parabola with a vertical axis_, the vertex being
the muzzle of the gun. Now, since each particle of the jet moves in
the same curve, this jet C takes the parabolic form. In every parabola
there is a certain point called the focus, and the distance from the
vertex to the focus fixes the magnitude of a parabola in much the same
manner as the distance from the center to the circumference fixes the
magnitude of a circle.
Now it can be proved that the focus B is as much below as the surface
of the water is above the orifice. Accordingly, if water issues through
orifices which are small in comparison with the contents of the
vessel, the jets from orifices at different depths below the surface
take different forms, as shown at D. If these curves are traced on
paper held behind the jet, then, knowing the horizontal distance and
the vertical height, it is easy to demonstrate that the jet forms a
parabola.
_Quantity of Efflux._—If we suppose the bottom of a vessel containing
water to be thin, and the orifice to be a small circle whose area is
A (see Fig. 100) where A B represents an orifice in the bottom of a
vessel.
[Illustration: FIG. 100.]
[Illustration: FIG. 101.]
Every particle above A B tries to pass out of the vessel, at once and
in so doing exerts a pressure on those nearest. Those that issue near
A and B exert pressures in the directions M M and N N; those near the
center of the orifice in the direction R Q, those in the intermediate
parts in the directions P Q, P Q. In consequence, the water within the
space P Q P is unable to escape, and that which does escape, instead of
assuming a cylindrical form, at first contracts, and takes the form of
a truncated cone.
It is found that the escaping jet continues to contract until at a
distance from the orifice about equal to the diameter of the orifice;
this part of the jet is called the _vena contracta_ or contracted vein,
as explained on a previous page.
_Influence of tubes on the quantity of efflux._—The result before
given has reference to an aperture in a thin wall. If a cylindrical or
conical efflux tube is fitted to the aperture, the amount of the flow
is considerably increased. A short tube, whose length is from two to
three times its diameter, has been found to increase the actual efflux
per second to about 82 per cent. of the theoretical. In this case the
water on entering the tube forms a contracted vein, Fig. 101. just as
it would do on issuing freely into the air; but afterwards it expands,
and, in consequence of the adhesion of the water to the interior
surface of the tube, has, on leaving the tube, a section greater than
that of the contracted vein. The contraction of the jet within the tube
causes a partial vacuum shown in black in the figure.
Now, if an aperture is made in the tube, near the point of greatest
contraction, and is carefully fitted with a vertical tube, the lower
end of which dips into water, Fig. 101, it is found that water rises
in the vertical tube, thereby proving conclusively the formation of a
partial vacuum.
If the nozzle has the form of a conic frustum whose larger end is at
the aperture, the efflux in a second may be raised to 92 per cent.,
provided the dimensions are properly chosen. If the smaller end of a
frustum of a cone of suitable dimensions be fitted to the orifice, the
efflux may be still further increased, which will fall very little
short of the theoretical amount.
[Illustration: FIG. 102.]
_Velocities of streams._—The velocity of streams varies greatly. The
slower flow of rivers has a velocity of less than three feet per
second, and the more rapid, as much as six feet per second, which gives
respectively about two and four miles per hour. The velocities vary in
different parts of the same transverse section of a stream, for the air
upon the surface of the water, as well also as the solid bottom of the
stream, has a certain effect in retarding the current. The velocity is
found to be greatest in the middle, where the water is deepest, Fig.
102, somewhere in _m_, below the surface; then it decreases with the
depth towards the sides, being least at _a_ and _b_.
[Illustration: FIG. 103.]
_Appearance of the surface during a discharge._—A vessel containing a
liquid, discharging itself through an orifice, does not always preserve
a horizontal surface. When the vein issues from an orifice in the
bottom of a vessel, and the level of the liquid is near the orifice,
the liquid forms a whirlpool, Fig. 103. If the liquid has a rotary
movement, the funnel is formed sooner; if the orifice is at the side of
the vessel, there is a depression of the surface upon that side, above
the orifice, Fig. 104. These movements depend upon the form of the
vessel, the height of the liquid in it, and the dimensions and form of
the orifice.
[Illustration: FIG. 104.]
In order to verify many of the laws of hydraulics in an accurate
manner, it is necessary to maintain a uniform pressure on the escaping
liquid, thereby obtaining a constant velocity at the orifice. This may
be done in various ways, as by allowing the water to flow into the
vessel in a little larger quantity than can escape from the orifice,
the excess being discharged over the upper edge of the vessel; also by
means of the syphon.
_By suspending solid particles_, such as charred paper, pulverized
in the water, we render the currents that are formed visible. These
solid particles arrange themselves, in curved lines, towards and into
the orifice, as a center of attraction, Fig. 105. The particles in
immediate contact with the orifice, not moving so easily as those
within, must cause contraction; so, also, _we can see that gravity in
accelerating the velocity, must cause continual decrease in the section
of the jet_.
[Illustration: FIG. 105.]
_Upward jets of water._—As the velocity of a liquid escaping from
an orifice is the same as that which a body acquires, falling from
a height equal to the distance from the level of the liquid to the
orifice, a jet of water escaping from a horizontal opening upwards,
should theoretically reach the level of the liquid in the vessel.
But this never takes place, Fig. 106, because of—1st, the friction in
the conducting tubes destroying the velocity. 2nd, the resistance of
the air. 3rd, the returning water falling upon that which is rising.
The height of the jet is increased by having the orifices very small,
in comparison with the conducting tube; piercing them in a very thin
wall, and inclining the jet a little, thus avoiding the effect of the
returning water.
[Illustration: FIG. 106.]
_Height of the jet._—If a jet issuing from an orifice in a vertical
direction has the same velocity as a body would have which fell from
the surface of the liquid to that orifice, the jet ought to rise to
the level of the liquid. It does not, however, reach this; for the
particles which fall hinder it. But by inclining the jet at a small
angle with the vertical it reaches about 9/10 of the theoretical
height, the difference being due to friction and to the resistance of
the air.
_The quantities of water which issue from orifices of different areas
are very nearly proportional to the size of the orifice_, provided the
level remains constant, and this is true irrespective of the form of
the opening which may be round, square, or any other shape.
_Escape of liquids through short tubes._—We often place in an orifice,
to increase the flow, a short tube (called an adjutage) either
cylindrical or conical. If the vein pass through the tube without
adhering to it, the flow is not modified; if the vein adhere (the
liquid wetting the interior walls) the contracted part is dilated, and
the flow is increased.
In the last case, and with a cylindrical adjutage, its length not
being more than four times its diameter, the flow is augmented about
one-third. Conical pipes, converging towards the exterior, increase the
flow still more than the preceding, the flow and velocity of the vein
varying with the angle.
_Escape of liquids through long tubes._—When a liquid passes through
a long straight tube, the flow soon diminishes greatly in velocity,
because of the friction which takes place between the liquid particles
and the walls. If there be any bends or curves in the tube, it is still
further diminished by the same cause. The discharge is then much less
than it would be from an orifice in a thin wall, and therefore the tube
is generally inclined; the liquid then passes down this inclined plane,
or it is forced through by pressure, applied at the opposite end.
_Direction of the jet from lateral orifices._—From the principle of the
equal transmission of pressure, water issues from an orifice _in the
side of a vessel_ with the same velocity as from an aperture in the
bottom of a vessel at the same depth.