WINDSAILS are the vanes, generally four in number, which, being turned by the action of tho wind, give motion to the machinery of a mill. The wind being supposed to blow in a direction parallel to the axis about which the sails are to revolve, it is evident that the plane of each sail must have a certain inclination to that axis, or to the plane of the revolution, in order that a resolved part of the wind's force may .act in the latter plane perpendicularly to the radii or arms which carry the sails so as to turn them constantly in one direction about the axis. If the pressure of the wind on the sails, supposed to be at rest, were to be alone considered, the determination of the angle which the plane of each sail should make with a plane perpendicular to the axis, or to the direction of the wind, in order that the pressure might be a maximum, would be comparatively easy. For by the resolution of forces it is easily seen that the pressure perpendicular to the radii, and in the plane of their revolution, varies with the term sine 0 cos 0, where 0 is the angle which the sail, supposed to be a plane surface, makes with the wind or with the axis of revolution : and the differential of this quantity being made equal to zero, the value of 0 is found to be 54° 44' nearly.
But it is evident that the effect of the wind in giving a revolving motion to the radii must depend on its pressure, and also on the velocity of the surface against which it acts ; and the angle which the plane of the sail should make with the direction of the wind, when Its pressure on the sail in motion is a maximum, must be determined by an investigation similar to that which follows.
Lot A B, A'n', parallel to one another, represent the direction of the wind ; wax, w'n'x', also parallel to one another, be two positions of n section of time sail, which by the pressure of the wind is made to move so that n, me, are in a line perpendicular to An. Now if it be supposed that At° is the space described by a particle of air while a would move to b (or to n') in the same direction, or from /I to n' in a direction perpendicular to Au; the lines A's' and 1/n' will, respectively, represent the velocities of the wind and sail in directions parallel to A'n', while nn' will be the velocity of the sail in the direction of this last line. Draw A'C' perpendicularly to wx or nix', produced, and meeting time former line in C; then A'C' and cc' will he respectively the velocities of the wind and sail perpendicularly to the line wx or w'x', and con sequently A'C will be what is called the velocity of the wind in the sail. Therefore, the pressure of a fluid being proportional to the square of the velocity, the pressure of the wind in time direction A'c' will vary with A'C'; and this being resolved in the direction A'D or nn', will be expressed by A'C" COS CA n, or sin n'n1/. But A'R being coast meat, A'C varies with sin A'uc ; therefore the effective pressure of the wind will vary with A' no sin Im' n6.
Let time angle A'nn' be represented by a, nb' by 0; then gilt.= a-0, and the expression for the pressure becomes sin= (a 0) sin 0.
Making the differential of this expression equal to zero, and reducing, we have tan (a-0) = 2 tan 0, when the pressure is a maximum.
Draw n's v perpendicular to nc, au that lex and x T may respectively represent tan (a 0 )and tan 0; and let A' me, n a' be respectively repre sented by r and by r'; then ne=t cos 0, n's =s1 sln 0, xv (=2Wx)=2esin0,and n'T =3e'sin 0.
Again, draw Y z perpendicular to nn', or parallel to A's'; then 13'2 ( = Y cos = 3v' sin= 0, Y ( =13' Y sin no's) = sin 0 cos 0, and = sin': 0.
But by similar triangles, sz : zv :: En' : that is sin' A : 3v' sin 0 cos 0 :: : v; whence v-31, sin 0 cos A.
Multiplying each term by v, and for the first term, substituting its equivalent sin' 04- r: coal A, we have 0 sin' 0 + cos" 8 sin 0 cos 0; or simplifying, and dividing by sine A, we get cotan 0=3vv cotan 0, which reduced as a quadratic equation, with respect to 0, gives 3 rs( cotan 0 (=tan sex) = ;77, + TT.) The angle AB x will evidently depend upon the relation between e, the velocity of the sail, and v, the velocity of the wind : if V = o, or the sail is at rest, we should have tan ABS= ,/2, that is, the angle sex would, as above. be equal to 54' 44' nearly ; and when ti=r, the for mula gives se m=74°19' nearly. It follows that as the velocity of the revolution increases, the iodination of the section wx to the wind, or to the axis of rotation, should be increased. Since, therefore, the velocity of the sail continually increases from the axis to the extremity of the radius or arm which carries it, it is evident that the sail, instead of being a plane. ought to have a curved surface such that the incli nation of the section to the direction of the wind may increase with its distance from the axis conformably to the values which would be given by the above formula, the ratio between the velocity of the wind and sail at any given distance from the axis of rotation being known or assumed. It was observed by Mr. Smeaton that the velo cities of the sails at their extremities are often more than twice as great as that of the wind. From several experiments which were made on a great scale by the same engineer, it was found that the effect is very advantageous when the inclinations of the axis, or the direction of the wind, with a section of the sail taken perpendicularly to the revolving arm at different distances from the axis, were as in the ()Hewing table : .1t one-sixth of the length of the arm . . . 72' At one-third 11 11 7r' At one-half IP 11 . 72' Al two-thirds . . . . 74' At live-sixths . . . 77r And at the extremity . . . . 83' Mr. Smeaton found also that when each sail is broader at the further extremity than near the centre, the effect is greater than when it hats the form of a parallelogram ; and that the most advantageous breadth at the extremity is one-third of the length of the arm.
There is a certain limit to the quantity of sail which a windmill can carry with advantage ; and from Mr. Smeaton 's experiments it results that, when the surfaces of all the sails exceeds seven-eighths of the irca of the circle described by each arm in one revolution, the velocity is diminished ; probably from the want of sufficient openings by which the wind, after impact, may escape. Mr. Smeaton also found that the ratio between the velocities of windmill sails when unconnected with the machinery, and when loaded so as to produce the maximum effect, is variable ; but, in general, that ratio is as 3 to 2. The velocity of the sails when the effect is a maximum varies nearly with the velocity of the wind.
The form and position of the sails remaining the same, the load or resistance when a maximum, varies nearly with the square of the velocity of the wind ; and the maximum of resistance which sails of similar figures, and in similar positions, will overcome at a given dis tance from the centre of motion, will vary with the cube of the radius or arm of the sail.