§ 29. _Application of the Principle of the Conservation of Energy to
Cases of Stream Line Motion._--The external and internal work done on
a mass is equal to the change of kinetic energy produced. In many
hydraulic questions this principle is difficult to apply, because from
the complicated nature of the motion produced it is difficult to
estimate the total kinetic energy generated, and because in some cases
the internal work done in overcoming frictional or viscous resistances
cannot be ascertained; but in the case of stream line motion it
furnishes a simple and important result known as Bernoulli's theorem.
[Illustration: FIG. 25.]
Let AB (fig. 25) be any one elementary stream, in a steadily moving
fluid mass. Then, from the steadiness of the motion, AB is a fixed
path in space through which a stream of fluid is constantly flowing.
Let OO be the free surface and XX any horizontal datum line. Let
[omega] be the area of a normal cross section, v the velocity, p the
intensity of pressure, and z the elevation above XX, of the elementary
stream AB at A, and [omega]1, p1, v1, z1 the same quantities at B.
Suppose that in a short time t the mass of fluid initially occupying
AB comes to A´B´. Then AA´, BB´ are equal to vt, v1t, and the volumes
of fluid AA´, BB´ are the equal inflow and outflow = Qt = [omega]vt =
[omega]1v1t, in the given time. If we suppose the filament AB
surrounded by other filaments moving with not very different
velocities, the frictional or viscous resistance on its surface will
be small enough to be neglected, and if the fluid is incompressible no
internal work is done in change of volume. Then the work done by
external forces will be equal to the kinetic energy produced in the
time considered.
The normal pressures on the surface of the mass (excluding the ends A,
B) are at each point normal to the direction of motion, and do no
work. Hence the only external forces to be reckoned are gravity and
the pressures on the ends of the stream.
The work of gravity when AB falls to A´B´ is the same as that of
transferring AA´ to BB´; that is, GQt(z - z1). The work of the
pressures on the ends, reckoning that at B negative, because it is
opposite to the direction of motion, is (p[omega] × vt) - (p1[omega]1
× v1t) = Qt(p - p1). The change of kinetic energy in the time t is the
difference of the kinetic energy originally possessed by AA´ and that
finally acquired by BB´, for in the intermediate part A´B there is no
change of kinetic energy, in consequence of the steadiness of the
motion. But the mass of AA´ and BB´ is GQt/g, and the change of
kinetic energy is therefore (GQt/g) (v1²/2 - v²/2). Equating this to
the work done on the mass AB,
GQt(z - z1) + Qt(p - p1) = (GQt/g)(v1²/2 - v²/2).
Dividing by GQt and rearranging the terms,
v²/2g + p/G + z = v1²/2g + p1/G + z1; (1)
or, as A and B are any two points,
v²/2g + p/G + z = constant = H. (2)
Now v²/2g is the head due to the velocity v, p/G is the head
equivalent to the pressure, and z is the elevation above the datum
(see § 16). Hence the terms on the left are the total head due to
velocity, pressure, and elevation at a given cross section of the
filament, z is easily seen to be the work in foot-pounds which would
be done by 1 lb. of fluid falling to the datum line, and similarly p/G
and v²/2g are the quantities of work which would be done by 1 lb. of
fluid due to the pressure p and velocity v. The expression on the left
of the equation is, therefore, the total energy of the stream at the
section considered, per lb. of fluid, estimated with reference to the
datum line XX. Hence we see that in stream line motion, under
the restrictions named above, the total energy per lb. of fluid is
uniformly distributed along the stream line. If the free surface of
the fluid OO is taken as the datum, and -h, -h1 are the depths of A
and B measured down from the free surface, the equation takes the form
v²/2g + p/G - h = v1²/2g + p1/G - h1; (3)
or generally
v²/2g + p/G - h = constant. (3a)
[Illustration: FIG. 26.]
§ 30. _Second Form of the Theorem of Bernoulli._--Suppose at the two
sections A, B (fig. 26) of an elementary stream small vertical pipes
are introduced, which may be termed pressure columns (§ 8), having
their lower ends accurately parallel to the direction of flow. In such
tubes the water will rise to heights corresponding to the pressures at
A and B. Hence b = p/G, and b´ = p1/G. Consequently the tops of the
pressure columns A´ and B´ will be at total heights b + c = p/G + z
and b´ + c´ = p1/G + z1 above the datum line XX. The difference of
level of the pressure column tops, or the fall of free surface level
between A and B, is therefore
[xi] = (p - p1)/G + (z - z1);
and this by equation (1), § 29 is (v1² - v²)/2g. That is, the fall of
free, surface level between two sections is equal to the difference of
the heights due to the velocities at the sections. The line A´B´ is
sometimes called the line of hydraulic gradient, though this term is
also used in cases where friction needs to be taken into account. It
is the line the height of which above datum is the sum of the
elevation and pressure head at that point, and it falls below a
horizontal line A´´B´´ drawn at H ft. above XX by the quantities a =
v²/2g and a´ = v1²/2g, when friction is absent.
§ 31. _Illustrations of the Theorem of Bernoulli._ In a lecture to the
mechanical section of the British Association in 1875, W. Froude gave
some experimental illustrations of the principle of Bernoulli. He
remarked that it was a common but erroneous impression that a fluid
exercises in a contracting pipe A (fig. 27) an excess of pressure
against the entire converging surface which it meets, and that,
conversely, as it enters an enlargement B, a relief of pressure is
experienced by the entire diverging surface of the pipe. Further it is
commonly assumed that when passing through a contraction C, there is
in the narrow neck an excess of pressure due to the squeezing together
of the liquid at that point. These impressions are in no respect
correct; the pressure is smaller as the section of the pipe is smaller
and conversely.
[Illustration: FIG. 27.]
Fig. 28 shows a pipe so formed that a contraction is followed by an
enlargement, and fig. 29 one in which an enlargement is followed by a
contraction. The vertical pressure columns show the decrease of
pressure at the contraction and increase of pressure at the
enlargement. The line abc in both figures shows the variation of free
surface level, supposing the pipe frictionless. In actual pipes,
however, work is expended in friction against the pipe; the total head
diminishes in proceeding along the pipe, and the free surface level is
a line such as ab1c1, falling below abc.
Froude further pointed out that, if a pipe contracts and enlarges
again to the same size, the resultant pressure on the converging part
exactly balances the resultant pressure on the diverging part so that
there is no tendency to move the pipe bodily when water flows through
it. Thus the conical part AB (fig. 30) presents the same projected
surface as HI, and the pressures parallel to the axis of the pipe,
normal to these projected surfaces, balance each other. Similarly the
pressures on BC, CD balance those on GH, EG. In the same way, in any
combination of enlargements and contractions, a balance of pressures,
due to the flow of liquid parallel to the axis of the pipe, will be
found, provided the sectional area and direction of the ends are the
same.
[Illustration: FIG. 28.]
[Illustration: FIG. 29.]
The following experiment is interesting. Two cisterns provided with
converging pipes were placed so that the jet from one was exactly
opposite the entrance to the other. The cisterns being filled very
nearly to the same level, the jet from the left-hand cistern A entered
the right-hand cistern B (fig. 31), shooting across the free space
between them without any waste, except that due to indirectness of aim
and want of exact correspondence in the form of the orifices. In the
actual experiment there was 18 in. of head in the right and 20½ in. of
head in the left-hand cistern, so that about 2½ in. were wasted in
friction. It will be seen that in the open space between the orifices
there was no pressure, except the atmospheric pressure acting
uniformly throughout the system.
[Illustration: FIG. 30.]
[Illustration: FIG. 31.]
§ 32. _Venturi Meter._--An ingenious application of the variation of
pressure and velocity in a converging and diverging pipe has been made
by Clemens Herschel in the construction of what he terms a Venturi
Meter for measuring the flow in water mains. Suppose that, as in fig.
32, a contraction is made in a water main, the change of section being
gradual to avoid the production of eddies. The ratio [rho] of the
cross sections at A and B, that is at inlet and throat, is in actual
meters 5 to 1 to 20 to 1, and is very carefully determined by the
maker of the meter. Then, if v and u are the velocities at A and B, u
= [rho]v. Let pressure pipes be introduced at A, B and C, and let H1,
H, H2 be the pressure heads at those points. Since the velocity at B
is greater than at A the pressure will be less. Neglecting friction
H1 + v²/2g = H + u²/2g,
H1 - H = (u² - v²)/2g = ([rho]² - 1)v²/2g.