wheel is kept face to the wind by a rudder in small mills; in large
mills a subsidiary fan and gear are used. Fig. 2 shows a large mill
of this kind, erected in a similar manner to a tower mill. The tower
is a framework of iron, and carries a revolving cap, on which the
wind shaft is fixed. Behind is the subsidiary fan with its gearing
Fig. 2.—Warner's Annular Sail Windmill.
acting on a toothed
wheel fixed to the cap.
It is important that a wind-mill should control itself so that it works efficiently in moderately strong winds and at the same time runs in very light winds, which are much more prevalent. It should also, by reefing or otherwise, secure safety in storms.
Table I. gives the mean velocity of the wind in miles per hour for an inland station, Kew, and a very exposed station, Scilly, for each month during the period 1890-1899.
The pressure of the wind on a plane normal to its direction, composed partly of an excess front pressure and negative back pressure, is given by the relation
p = 0.003 v2,
where p is in pounds per square foot and v the velocity of the wind in miles per hour. It varies a little with the form and size of the surface, but for the present purpose this variation may be disregarded. (See experiments by Dr Stanton at the National Physical Laboratory, Proc. Inst. Civ. Eng. 156, p. 78.). For velocities of 5, 10 and 20 m. per hour the pressures on a plane normal to the wind would be about 0.075, 0.3 and 1.2 ℔ per sq. ft. respectively, and these may be taken to be ordinary working velocities for windmills. In storms the pressures are much greater, and must be reckoned with in considering the stability of the mill. A favourable wind velocity for windmills is 15 m. per hour.
| Table I. | ||||||||||||||||||||||||||
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Pressure on Surfaces oblique to the Wind.—Let fig. 3 represent a plane at rest on which a wind current impinges in the direction YY, making an angle θ with the normal Oa to the plane. Then the pressure n normal to the plane is given very approximately by Duchemin's rule
n = p2 cos θ1 + cos2 θ ℔ per sq. ft.
where p is the pressure in pounds per square foot on a plane struck normally by the same wind.
In fig. 3 let AB be part of a windmill sail or vane at rest, XX
being the plane of rotation and YY the direction of the wind. The
Fig. 3.
angle θ is termed the
weather of the sail. This
is generally a constant
angle for the sail, but in
some cases varies from a
small angle at the outer
end to a larger angle near
the axis of rotation. In
mills of the European type,
θ − 12° to 18°, and the
speed of the tips of the
sails is 212 to 3 times the
velocity of the wind. In
mills of the American
type, θ = 28° to 40°, and
the speed of the tips of the
vanes is 34 to 1 time that of the wind. Then if Oa = n be the normal
pressure on the sail or vane per square foot, ba = t is the effective
component of pressure in the direction of rotation and
t = n sin θ = p2 sin θ cos θ1 + cos2 θ
When the sail is rotating in a plane at right angles to the wind direction the conditions are more complicated. In fig. 4 let XX be the plane of rotation of the vane and YY the direction of the wind. Let Oa be the normal to the vane, θ being the weather of the vane. Let Ov = v be the velocity of the wind, Ou = u the velocity of the vane. Completing the parallelogram, Ovr=vr is the velocity and direction of the wind relatively to the vane.
vr = √ (v2+u2) = v sec φ,
tan φ = u/v,
and the angle between the relative direction of wind and normal to the vane is θ+φ. It is clear that θ+φ cannot be greater than 90°, or the vane would press on the wind instead of the wind on the vane. Substituting these values in the equations already given, the normal pressure on the oblique moving vane is
n = .003 v2 sec2 φ2 cos(θ+φ)1+cos2(θ+φ)
The component of this pressure in the direction of motion of the vane is
t = .003 v2 sec2 φ2 sin(θ+φ) cos(θ+φ)1+cos2(θ+φ)
and the work done in driving the vane is
tu = tv tan φ
= .003 v3 sec2 φ tan φ2 sin(θ+φ) cos(θ+φ)1+cos2(θ+φ)
foot ℔ per sq. ft. of vane per sec., where v is taken in miles per hour.
Fig. 4.
For such angles and
velocities as are
usual in windmills
this would give for a
square foot of vane,
near the tip about
0.003 v2 ft. ℔ per
sec. But parts of
the vane or sail
nearer the axis of
rotation are less
effective, and there
are mechanical friction
and other
causes of inefficiency.
An old rule
based on experiments
by Coulomb on mills of the European type gave for the
average effective work in foot ℔ per sec. per sq. ft. of sail
W = 0.0011 v3
| Table II.—In 150 Working Hours. | ||||||||||||||||||||||||||||||||||||||||||
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| I. Goold Shapley and Muir, Ontario; wheel 16 ft. diameter, 18 vanes, 131 sq. ft. area (first prize). II. Thomas & Son (second prize). III. J. W. Titt. IV. R. Warner. V. J. W. Titt. VI. H. Sykes. |
