THE 

 LONDON, EDINBURGH, and DUBLIN 



PHILOSOPHICAL MAGAZINE 



AND 



JOURNAL OF SCIENCE. 



[FIFTH SERIES.] 



OCTOBER 1886, 



XXXVII. Turbines. By J. Lester Woodbridge, B.S., M.E* 



I PROPOSE to discuss the action of turbines in general. 

 Consider first the water after it has entered the wheel 

 and is passing along its vanes. Conceive the water to be 

 divided into an infinite number of filaments by vanes similar 

 to those of the wheel, but subjected to the condition that, at 

 each point, their width, ac (fig. 1), measured on the arc whose 

 centre is 0, shall subtend at the centre a constant angle dd. 

 Conceive each filament to be divided into small prisms whose 

 bases are represented by the shaded areas a/We'd', d'dd'd", and 

 abed, by vertical planes normal to the vanes, making the 

 divisions ae, ef, intercepted on the radius by circles passing 

 through the consecutive vertices on the same vane a!, d f , d", &c, 

 equal. The variable height of a prism represent by #, and let 

 p be the variable distance from the centre. 

 Then dp = ae,ef,&c. 



pd6dp=abcd, &c.=area of the base of an infinitesimal 

 prism ; 

 xpdQdp-=- volume of infinitesimal prism ; 

 xhpdddp = m=-thQ mass of prism, 8 being its density ; 



7 = san = angle between the normal to the vane at 

 any point p, and the radius Oa prolonged 

 through that point ; 

 v — the velocity of a particle along the vane at p ; 

 co = the uniform angular velocity of the wheel ; and 

 jD = the pressure of the water at the point p. 



* Communicated by De Volson Wood, Professor of Engineering in 

 Stevens Institute of Technolog}', Hoboken, N.J. 



Phil. Mag. S. 5. Vol. 22. No. 137. Oct. 188G. Y 



