THKORETK'AL AND MEASURED PUMPIN(; POWER OF WINDMILLS. 97 



the wind should increase from the inner end of the fan to the outer 

 end; hence for maximum power the fans should be warped surfaces. 



Case I. 



The fans as in Fig. 2 with x constant. This is the form of fan 

 used in nearly all American steel mills. 



Substituting these values in (6) we have 

 Y^^VcosSxR^ARI 



,L'=K~ ~ ^ cos xl c2sin2xR2/\R— 2^c sin x cos x R^ AR 



Making A R infinitesimal=dR and integrating R between limits 

 R„ and R„ and dL' between corresponding limits we have 



/^L r V r R 3 T? 3 V 



L'= ( dL'-=K- b^-^cos xc3 sin^x " ^ „ ° 2-i'sin x cos x 



Rn ^-Rp^ /v„\- , R„s-Ron , ^ 



-4i^;^-(c)^^^^^^-lR^J ^-^ 



The ratio of =— ^ is nearly constant for different sizes of wheels. 



Rn 



Letting this ratio be t and substituting it in ( 1 1 ) and multipl3ang by 

 n, the number of fans in wheel, we have for the power of the mill 



L=K — nD„ - cos X C'^R,, sm-x — 2 — ( ism x cos x 



gc L3 cV4/ 



Example: — Six-foot steel mill No. 7, Table II. 



n=:i2, b„ = ^ , ---=.311 (from Fig. 7). x^=ss'\ — ^0021. 

 ' 12 c o / y g 



Substituting these values in (12) we have 



Le=KX.oo2i Xi2X|lX.3iiX-5736Xc^[.326x.67i— 2X-311X 



.2485X-8i9 + .3ii2x.20X-329] 

 =.oi6Kc3 [.2187 — .0725 + . 0064] 



=.0024X0'' (13) 



The value of the coefficient — K is difficidt to determine. Pro- 

 fessor Weisbach assumed that the suction on the leeward side of a 

 fan is one-half the pressure on the windward side. This would 

 make K^-| for a plane fan. The writer has found from experiment 

 that the normal wind pressure on the concave side of a smooth iron 

 surface such as is used for windmill fans is 1.15 times that on a 



