324 Mr. J. L. Woodbridge on Turbines. 



in regard to which he states that for w a minimum cop 2 must 

 equal v^ which is not generally true, and is true only when 

 7 2 = 0, or when v 2 cop 2 and ((op 2 — v 2 y happen to be a minimum 

 together. The value of a> in equation (24) will not in general 

 satisfy Rankine's condition 



cop 2 =v 2 cos y 2 , . . . . . (29) 

 nor Weisbach's 



(op 2 = v 2 (29a) 



Substituting in equation (24) Rankine's condition yi = 90°, 

 and making r=— , we find 



92 



to *= ^ H f l-^- •(l-r »)«-cos«7 8 (l-2»*)1 

 H2 l-2r 2 L ^/(l-r 2 ) 2 -cos 2 72 (l-2r 2 ) J' V ; 

 Substituting this in (15a) gives 



v <2 cosy 2 = G>p 2 [l-r 2 + s/ (l-/' 2 ) 2 -cos 2 y 3 (l-2r 2 )]. (31) 



This satisfies equation (29) only when r 2 = %, and (29a) 

 only when 7 2 = or r=l. The latter condition is that of 

 a parallel flow wheel, or of an infinitely narrow wheel, in 

 which case we have 



(op 2 =v 2 = vVH.. 



These in (17) give for the efficiency 



E = cos7 2 , 



which always exceeds the value given by Rankine, when r = 1, 



E 



2 cos 2 7 2 

 1 + cos 2 7 2 ' 



except when 7 2 = 0, when both become unity, but the work 

 done will be zero. 



The pressure in the wheel may be found by integrating 

 equation (3) between initial and general limits, and elimi- 

 nating p x and v by means of equations (12) and (9), giving 



|= ft ,y-2 6 ,v+| 2 +%D+2« 1 ^ 1 ™ S ?.-V$^, (32) 



which may be discussed for the various conditions to which 

 the wheel is subjected. 



