THEORY OF TURBINE DESIGN 539 



If ft = 90, w 2 = w 2 , and since = - 



tan y = -^ \ tan a. (13) 



&3 ff 



If / is constant this gives : 



tan y = tan o. (14) 



* 



And if = H-, 



tan y = ?i tan a. (1 5) 



Change of Pressure through Wheel. Since the motion of a particle of 

 ,'ater at any point in the wheel may be compounded of its motion in a 

 arced vortex with angular velocity , and of its motion parallel to the 

 rtieel vanes with (variable) velocity v rt the difference of pressure at any 

 wo points in the wheel will be the algebraic sum of the differences 

 lecessary to produce these motions. 

 Thus, due to the vortex motion, 



yhile due to the flow between the vanes, 



P*" ~ p *" = * v *~ *'* feet. 



Summing these we have : 



Cotal difference of pressure at | _ p 2 PS _ 1 ^Ji_j^ i 3^ r a - 2^V 2 

 inlet and outlet. } ~ W 2 g~~ 2g 



But 3 v r = / 3 cosec y, and 2 v r = / 2 cosec ft, while w 3 = , 



. Pz Pa _ W 2 / ., 1 \ , y 3 2 cosec 2 y / 2 2 cosec 2 /8 nfi , 



W ~ 2 V n *) ' Zg 



[he pressure p, at any radius r between inlet and outlet may be obtained 

 f the angle 6 made by the vanes at this radius with the tangent to the 

 ;orresponding circle, and the velocity of flow,/, be known. 

 Here, if u be the velocity of the wheel at r, 



u = -~, while v r = / cosec 6, 



/ 2 cosec 2 / 2 2 cosec 2 ft 



r 



Ibs. per square foot. (17) 



