§ 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)
[Illustration: FIG. 58.]
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