Water-Wheels; Or, Hydraulic Motors (Paperback)

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1876 Excerpt: ...at the same time, to give the value of the unknown quantities. Adding equations (1), (2), and (5), member to member, there obtains or, considering (3) and (9), u v cos /3 = g H.... (10) The combination of equations (4) and (9) readily give b v r sin /3 = b ' u r sin y;.... (11) multiplying equations (7), (10), and (11), member by member, there obtains v' b r sin / 3 cos /3 = g H. b ' r" sin y; whence one of the unknown quantities v" = gl -5--r---.... (12) o r sin j9 cos /3 J Til "We have besides, from equation (10), u =---1--; whence 'u v n v cos/3, -rr 5 r' tan /3.. u = ffH-TT-n-..... (13) v b' r" sm y' K' and from equation (7) = Rb tan y J' sin 7 v' To obtain r', we will first make w' = u' in equation (6), which will give vn = 2 u" (1-cos y), and consequently from the value of u' (14), vn = 2 ff H 4 (1-cos 7).... (15). 5' sin 7 v /y v y Knowing u' we obtain w;', and, if w were required, we could easily obtain it by substituting in equation (5) the values of v and u. Thus, all the velocities may be considered as known; we could deduce from them the angular velocity u with which the turbine must move, when it is working with the conditions of the maximum effective delivery; we would then have practically _ u _ u' r r' The corresponding expense Q has for its value 2 b' w' r' sin y, or, substituting in place of w' the value of its equal u' from equation (14), Q = 2 V V b V V g H tan /3 sin 7, .... (16) a formula whose second member we should probably have to multiply by a number less than unity, in order to take into consideration the space occupied by the floats, and also to compensate for the influence of the losses of head neglected in the calculation. Let us now seek the three equations of condition to be satistied by th...