v_a = c_v[root](2gh) = [root]{2gh/(1 + c_r)}
c_v = [root]{1/(1 + c_r)} (5)
c_r = 1/c_v² - 1 (5a)
Thus if c_v = 0.97, then c_r = 0.0628. That is, for such an orifice
about 6¼% of the head is expended in overcoming frictional resistances
to flow.
[Illustration: FIG. 15.]
_Coefficient of Contraction--Sharp-edged Orifices in Plane
Surfaces._--When a jet issues from an aperture in a vessel, it may
either spring clear from the inner edge of the orifice as at a or b
(fig. 15), or it may adhere to the sides of the orifice as at c. The
former condition will be found if the orifice is bevelled outwards as
at a, so as to be sharp edged, and it will also occur generally for a
prismatic aperture like b, provided the thickness of the plate in
which the aperture is formed is less than the diameter of the jet. But
if the thickness is greater the condition shown at c will occur.
When the discharge occurs as at a or b, the filaments converging
towards the orifice continue to converge beyond it, so that the
section of the jet where the filaments have become parallel is smaller
than the section of the orifice. The inertia of the filaments opposes
sudden change of direction of motion at the edge of the orifice, and
the convergence continues for a distance of about half the diameter of
the orifice beyond it. Let [omega] be the area of the orifice, and
c_c[omega] the area of the jet at the point where convergence ceases;
then c_c is a coefficient to be determined experimentally for each
kind of orifice, called the _coefficient of contraction_. When the
orifice is a sharp-edged orifice in a plane surface, the value of c_c
is on the average 0.64, or the section of the jet is very nearly
five-eighths of the area of the orifice.
_Coefficient of Discharge._--In applying the general formula Q =
[omega]v to a stream, it is assumed that the filaments have a common
velocity v normal to the section [omega]. But if the jet contracts, it
is at the contracted section of the jet that the direction of motion
is normal to a transverse section of the jet. Hence the actual
discharge when contraction occurs is
Q_a = c_vv × c_c[omega] = c_c c_v[omega][root](2gh),
or simply, if c = c_vc_c,
Q_a = c[omega][root](2gh),
where c is called the _coefficient of discharge_. Thus for a
sharp-edged plane orifice c = 0.97 × 0.64 = 0.62.
[Illustration: FIG. 16.]
§ 18. _Experimental Determination of c_v, c_c, and c._--The
coefficient of contraction c_c is directly determined by measuring the
dimensions of the jet. For this purpose fixed screws of fine pitch
(fig. 16) are convenient. These are set to touch the jet, and then the
distance between them can be measured at leisure.
The coefficient of velocity is determined directly by measuring the
parabolic path of a horizontal jet.
Let OX, OY (fig. 17) be horizontal and vertical axes, the origin being
at the orifice. Let h be the head, and x, y the coordinates of a point
A on the parabolic path of the jet. If v_a is the velocity at the
orifice, and t the time in which a particle moves from O to A, then
x = v_a t; y = ½gt².
Eliminating t,
v_a = [root](gx²/2y).