Mr. J. L. Woodbridge on Turbines* 3T9 



the water at any point,, and 8Qp[cop— v cos 7] is the moment 

 of the momentum ; hence, integrating between limits for the 

 inner and outer rims, the moment exerted by the water on the 

 wheel equals the difference in its moment of momentum on 

 entering and leaving the ivheel. Thus Ave have deduced an 

 expression which some writers have made the basis of their 

 investigations. 



Let the values of the variables at the entrance of the wheel- 

 be 



pi, 7h v u p lr 

 and at exit be 



P2> 72? v 2 , ]?2- 



Then equations (3) and (5) become 



M = SQ[a)(p 1 2 — p 2 2 ) —piv x cos 7i + p 2 ^2 cos 72], (7) 



.*. U=Mo) = BQco[co(p i 2 — p 2 *) — p x v t cosy! +p 2 v 2 cos y 2 ]. (8) 



Equation (8) gives the work per second in terms of the 

 known quantities B, to, p l} y l} p 2 , y 2 , and the three quantities 

 Q, vi, v 2 as yet unknown. These three quantities are, how- 

 ever, connected by the condition that the quantity of water 

 flowing through, all the sections radially is constant. Calling 

 a 1 the entire area of all the orifices at the entrance of the wheel 

 ( = 27rp 1 x 1 ), and a 2 those at exit ( = 27rp 2 x 2 ), we have 



Q ) = aiVisinji = a 2 v 2 smy 2) .... (9) 



which reduces this number of unknown quantities to one. 



Equation (6) is the equation of the motion of the water in the 

 wheel. Besides the velocities v x and v 2 , it contains^ &ndp 2 > 

 Let 

 p a =the atmospheric pressure, 

 h =the mean depth of the wheel below the surface of 



the tail-race, 

 p t =Sgh = the pressure due to flooding in the tail-race ; 

 then 



p 2 =Pa+Pf 



The pressure p x where the water passes from the guide- 

 plates into the wheel is unknown. Another condition is 

 necessary, which may be found by considering the passage 

 of the water from the guide-plates into the wheel. In fig. 2 

 let A C be the tangent to the guide-plate at its extremity, V 

 the actual velocity of the water on leaving the guide-plate, 

 (op x = AD the velocity of the initial rim of the wheel ; then 



