. The steam-engine and other heat-motors. «, before it meets aresistance and an absolute velocity of V2 afterwards the loss of SUPERHEATED STEAM AND STEAM-TURBINES. 455 energy of the stream is w(vp-v22) ;y?-v#) This 2g 2g loss of energy may appear as the work of friction, useful work inmoving the vane against a resistance or both combined. Disregarding friction, let us determine the conditions of maxi-mum efficiency. It will evidently depend upon making V2 or bd,Figs. 236 and 237, a minimum. bd2 = ed2+eb2-2(ebxef). Disregarding friction, ab = bc = ed, bd2 = ab2+eb2-2{ebxej),ae2 = ab2+e
. The steam-engine and other heat-motors. «, before it meets aresistance and an absolute velocity of V2 afterwards the loss of SUPERHEATED STEAM AND STEAM-TURBINES. 455 energy of the stream is w(vp-v22) ;y?-v#) This 2g 2g loss of energy may appear as the work of friction, useful work inmoving the vane against a resistance or both combined. Disregarding friction, let us determine the conditions of maxi-mum efficiency. It will evidently depend upon making V2 or bd,Figs. 236 and 237, a minimum. bd2 = ed2+eb2-2(ebxef). Disregarding friction, ab = bc = ed, bd2 = ab2+eb2-2{ebxej),ae2 = ab2+eb2+2(ebxbg),bd2 = ae2 -2(eb xbg) -2{eb xef),bd2 = V22 = V? -2v(Vi cos a +Vi cos ft),V? = V2 -2vVi(cos a +cos ft). Maximum Efficiency under Various Conditions. — 1. If thestream is turned tlirough 180° both a and ft will be zero, hencecosa = cos/? = l. V2 = 0 and v=Vi=¥r. For the highest theoretical efficiency the stream should beturned through 180° and the velocity of the vane should be one-half the velocity of the entering 2.: a, andmum. Fig. 237. As a rule, it is not feasible to have a=ft=0. When Ve, ft are fixed, V2 has a minimum value when (vV{) is a maxi- If Vs and a are fixed in amount, a circular arc can be~ 456 THE STEAM-ENGINE AND OTHER HEAT-MOTORS. passed through a, b, and e, since we have given a chord and the anglesubtended by the chord and as Vi must be less than V£. Let abe vV• sin abe the circular arc and b the varying point on it. Now —%— is the area of the triangle abe and so is ae multiplied by the altitude. Since sin a: is a constant, anything that increases the area of the triangle increases the value of vVi. The altitude perpendicular to ae is a maximum when b is the middle point of <xthe arc abe. Hence vV{ is a maximum when v=Vi or ^=iy Therefore, when Ve, a, and ft are fixed, the best economy occurs a V£ when the speed of the blade, v, is such that v cos-^=-^-. 3. If V, is fixed in direction, and we know that ft=a, butwe do not k
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