96 



KANSAS UNIVERSITY QUARTERLY. 



The power AL^ of /\s is equal to the force in (4) muUipHed by 

 V, the velocity of i\s. Hence 



X 



/\Li= '\P COS X v=K Asv(c sin x — vcos x)2cos x. . (6) 



i — 1 , — ^ . — 



Equation (6) can be put in the form 



AL^ = K^c3AsB (7) 



V 



in which B=a cos x(sin x — a cos x)^ and a^ — . 



c 



Table I gives values of B for values a and x. 



TABLE L 



From (7) we see that for a given wind velocity c and surface 

 A s the power varies with B, and, from Table I, we see for any 

 value of a, the proper value of x for maximuai power, h^or cx- 



V 



ample: If a = — =1.8, x=:7o° approxunatelw 

 c 



The relation between x and a for maximum power is found as 



follows: Differentiating (6) with respect to x we have 



d L' T^ r , ^ ^ „ . ... . ., ., 



:K — c^Asa[2cos- X sin X — siu" x-| 4a cos X sm-x — 2a cos'^x 



..(8) 

 ••(9) 



dx 



• — 3a2cos-xsin x] =0 



Simplifying (8) we have 



— tan3x-(-4a tan~xA(2 — 3a-)tan x — 2a=o, 

 Solving (9) we have 



tan. x=::a and |adr] (fa)^ + 2. 

 The first of these values makes AL'=-0 and gives a minimum 

 value; of the other two fa-f] ■'\'^a)^-^2 gives the greatest value of 

 AL'. Hence the relation between x and a for maximum power is 



tan x=:|a+i (fa)'--} 2 ( lo) 



By substituting for a any desired value in (10) the corresponding 

 value of x for maximum power is found. 



V 



For any given value of c the ratio increases as the radius of 



the wheel increases, and. since an angle increases with its tangent, 

 we see that the angle which a fan makes with the direction of 



