Page:Encyclopædia Britannica, Ninth Edition, v. 12.djvu/503

487 HYDRAULICS.] HYDROMECHANICS 487 to abrupt changes in the velocity of the stream producing eddies. The kinetic energy of these is deducted from the general energy of translation, and practically wasted. Sudden Enlargement of Sec tion. Suppose a pipe enlarges in section from an area o&amp;gt; to an, area w t (fig. 86) ; then or, if the section is circular, d The head lost at the abrupt change of velocity has already been shown to be the head due to the relative velocity ot the two parts of the stream. Hence head lost J V rr^- , 20 if & is put for the expression in brackets. o&amp;gt; 1-1 1-2 1-3 1-4 1-5 1-6 17 1-8 1-9 | 2-0 2-5 3-0 3-5 4-0 5-0 6-0 7-0 8-0 rf,_ 1-05 1-10 1-14 1-18 1-22 1-26 1-30 T34 1-38 1-41 1-58 1-73 1-87 2-00 2-24 2-45 2-65 2-83 fc- 01 04 09 16 25 36 49 64 81 1-00 2 25 4-00 6-25 9-00 16-00 25-00 36-0 49-0 Abrupt Contraction of Section. When water passes from a larger to a smaller section, as in figs. 87, 88, a contraction is formed, and the contracted stream abruptly expands to fill the section of the pipe. Let o&amp;gt; be the section and v the velocity of the stream at Ib. At aa the section will be c c &amp;lt;a, and the velocity v =, where c e is the coefficient of contraction. Then the head lost is and, if c c is taken - 64, 2? 2gr (2). The value of the coefficient of contraction for this case is, however, not well ascertained, and the result is somewhat modified by fric tion. For water entering a cylindrical, not bellmouthed, pipe from a reservoir of indefinitely large size, experiment gives f)a = 505 ....... (3). 20 If there is a diaphragm at the mouth of the pipe as in fig. 88, let &amp;lt;a l be the area of this orifice. Then the area of the contracted stream is c c o&amp;gt; v and the head lost is if is put for ( -?- Weisbach has found experimentally the following values of the coefficient, when the stream approaching the orifice was consider ably larger than the orifice : Uj 0-1 2 03 0-4 0-5 0-6 0-7 0-8 9 1-0 Ct- Gie 614 612 610 607 605 603 601 598 596 &amp;lt;i= 231-7 50-99 19-78 9-612 5-2.-&amp;gt;&amp;lt;; 3-077 1-876 1 -169 0-734 0-480 When a diaphragm was placed in a tube of uniform section (fig. 89), Fig. 89. the following values were obtained, a) x being the area of the orifice, and u that of the pipe : ^! = o-i 0-2 0-3 0-4 0-5 0-6 0-7 0-8 0-9 1-n c,= 624 632 643 659 081 712 755 813 892 i-oo = 225-9 47-77 30-83 7-801 1 753 1-796 797 290 o&amp;lt;;&amp;lt;&amp;gt; 000 . Weisbach considers the loss of head at elbows (fig. 90) to be due to a contraction formed by the stream. From experiments with a pipe l inches dia meter, he found the loss of head 6 = 0-9457 sin 2 + 2-047 sin 4 4&amp;gt; = & = 20 0-046 40 0-139 60 0-364 80 0-740 90 0-984 100 1-200 110 1-556 120 1-861 130 2-158 140 2-431 Hence at a right-angled elbow the whole head due to the velocity very nearly is lost. Sends. Weisbach traces the loss of head at curved bends to a similar cause to that at elbows, but the coefficients for bends are not very satisfactorily ascertained. Weisbach obtained for the loss of head at a bend in a pipe of circular section d (6); where d is the diameter of the pipe and p the radius of curvature of the bend. For bends with rectangular cross sections = 0-124 + 3-104 where $ is the length of the side of the section parallel to the radius of curvature p. d _ o-i 0-2 0-3 0-4 0-5 0-6 0-7 0-8 0-9 1-0 &amp;lt;= 0-131 &quot;138 158 206 294 440 661 977 1-408 1-978 s o-i 0&quot;2 0-3 0-4 0-5 0-6 0-7 0-8 0-9 i-o 2p ft- 124 135 180 250 398 643 1-015 1-546 2-271 3 228 J r ahrn, Cocks, and Sluices, These produce a contraction of the water-stream, similar to that for an abrupt diminution of section already discussed. The loss of head may be taken as before to be k-f. (?); where v is the velocity in the pipe beyond the valve and &amp;gt; a coefficient determined by ex periment. The following are Weisbach s re- Sluice in Pipe of Rectangular Section (fig. 91). Section at sluice = u&amp;gt;! in pipe =. &quot;&quot;= 1-0 J 0-9 0-8 0-7 06 0-5 04 0-3 0-2 to 1 s r== O OO ! -09 39 95 2 4-02 S-12 17-8 44-5