[Illustration: FIG. 162.]
The work done by the jet on the vane is Pu = (G/g)btu(v - u)²(1 - cos
[phi]), which is a maximum when u = 1/3 v. This result can also be
obtained by considering that the work done on the plane must be equal
to the energy lost by the water, when friction is neglected.
If [phi] = 180°, cos [phi] = -1, 1 - cos [phi] = 2; then P =
2(G/g)bt(v - u)², the same result as for a concave cup.
[Illustration: FIG. 163.]
§ 161. _Position which a Movable Plane takes in Flowing Water._--When
a rectangular plane, movable about an axis parallel to one of its
sides, is placed in an indefinite current of fluid, it takes a
position such that the resultant of the normal pressures on the two
sides of the axis passes through the axis. If, therefore, planes
pivoted so that the ratio a/b (fig. 163) is varied are placed in
water, and the angle they make with the direction of the stream is
observed, the position of the resultant of the pressures on the plane
is determined for different angular positions. Experiments of this
kind have been made by Hagen. Some of his results are given in the
following table:--
+-----------+-------------+--------------+
| |Larger plane.|Smaller Plane.|
+-----------+-------------+--------------+
| a/b = 1.0 |[phi] = ... |[phi] = 90° |
| 0.9 | 75° | 72½° |
| 0.8 | 60° | 57° |
| 0.7 | 48° | 43° |
| 0.6 | 25° | 29° |
| 0.5 | 13° | 13° |
| 0.4 | 8° | 6½° |
| 0.3 | 6° | .. |
| 0.2 | 4° | .. |
+-----------+-------------+--------------+
§ 162. _Direct Action distinguished from Reaction_ (Rankine, _Steam
Engine_, § 147).
The pressure which a jet exerts on a vane can be distinguished into
two parts, viz.:--
(1) The pressure arising from changing the direct component of the
velocity of the water into the velocity of the vane. In fig. 153, §
154, ab cos bae is the direct component of the water's velocity, or
component in the direction of motion of vane. This is changed into the
velocity ae of the vane. The pressure due to direct impulse is then
P1 = GQ(ab cos bae - ae)/g.
For a flat vane moving normally, this direct action is the only action
producing pressure on the vane.
(2) The term reaction is applied to the additional action due to the
direction and velocity with which the water glances off the vane. It
is this which is diminished by the friction between the water and the
vane. In Case 2, § 160, the direct pressure is
P1 = Gbt(v - u)²/g.