540 HYDRAULICS AND ITS APPLICATIONS 



If the velocity of flow is constant / 2 = /, and this expression simplifies 

 to : 



W F o ( / r \ 2 ) "1 



p = j> 2 o v ^ 1 * ~~ ( J " / i + / 2 ( cosec 2 # cose c W . 



a Q I*, \ '/ 2 / - J 



As every term in this expression is known, the pressure at any point 

 And therefore the whole pressure on that portion of the runner whicl 

 carries the vanes, may be determined. 



An easier graphical method is, however, indicated in Art. 145. 



The pressure over the face of the runner inside the vanes = p^ Ibs. pei 

 square foot, so that the total pressure over this face may be determined 

 Owing to leakage past the outer periphery of the runner the pressure or 

 the rear face may, however, amount to as much as_p 2 Ibs. per square foot, anc 

 owing to the large difference of pressure thus produced on the two faces 

 the end thrust on the shaft may become excessive. Various methods o 

 balancing this end-thrust have been adopted, and have been illustrated ii 

 the preceding chapter. 



Further consideration of these methods will be postponed to Art. 183 

 where the similar problem of balancing the end thrust on the spindle of i 

 centrifugal pump is considered in some detail. 



Summary. Collecting the more important of the results so far obtaine< 

 we have in the case of an inward radial flow turbine, working withou 

 shock at entrance and rejecting its discharge water without any tangentia 

 velocity, and therefore (neglecting the effect of other losses) working at its 

 maximum efficiency, 



/i\ , _ / i _ tan a \ 



^ ' a 2 ~\ tan 8 J 



J O f, TT ' 



(i \ I n ji 



i tan a \ I 



} \. / n n tan a . / n Da . Y- 



tan B J V 2 2 - -f ^ tan a 



v tan 6 * \ b a J 



(2) rt / 2 = w% tan a feet per second. 



(8) tt tan 7 = n ^-y- 2 tan a 



(4) a / 3 = n ^-/ 2 feet per second. 



(5). p = P2 - - ' v/ 2 2 | 1 - ( ) 2 I + / 2 cosec 2 -/ 2 2 cosec 2 /3 



*/ ^ 



Ibs. per square foot. 



2 



(6) Hydraulic efficiency 77 = -. ' , v -^r 



2 _(- ( Y n tan a ) X -- 



- tan a 



~ 



