486 HYDROMECHANICS [HYDRAULICS. a = 001262 ai = -000833 = -00206 j g 1 = -00001042 74. Scraping Water Mains. The influence of the condition of the surface of a pipe on the friction is shown by various facts known to the engineers of waterworks. Pipes are very often heated and dipped in pitch, which gives them a smooth hard surface and protects them from oxidation. Such pipes are known to give a discharge larger than that calculated by the ordinary formula;. In pipes which convey certain kinds of water oxidation proceeds rapidly, and the discharge is very perceptibly and sometimes very greatly diminished. In a main laid at Torquay the discharge diminished from this cause more than 50 per cent. , and the supply became insufficient for the town. Mr Appold suggested an apparatus for scraping the interior of the pipe, and this was constructed and tried under the direction of Mr Fronde. It was found that by scraping the interior of the pipe the discharge was increased 56 per cent. The scraping requires to be repeated at intervals. After each scraping the discharge dimin ishes rather rapidly by 10 per cent, and afterwards more slowly, the diminution in a year being about 25 per cent. Fig. 83 shows a scraper for water mains, similar to Mr Appold s but modified in details, as constructed by the Glenfield Company, at Kilmarnock. A is a longitudinal section of the pipe, showing the scraper in place ; B is an end view of the plungers, and C, I) sec tions of the boxes placed at intervals on the main for introducing or withdrawing the scraper. The apparatus consists of two plungers, packed with leather so as to fit the main pretty closely. On the for the successive portions, and let I, d, v, and i be corresponding quantities for the equivalent uniform main B. The total loss of head in A due to friction is and in the uniform main If the mains are equivalent, as defined above, V14Z V i.v>V4*8 +
- 2gr d *yg d 1 ^2g d 2
But, since the discharge is the same for all portions, Also suppose that may be treated as constant for all the pipes, Then I d 4 I d* I d d^ d 1 o? 2 4 d. 2 7 CL , a , spindle of these plungers are fixed eight steel scraping blades, with curved scraping edges fitting the surface of the main. The appa ratus is placed in the main by removing the cover from one of the boxes shown at C, D. The cover is then replaced, water pressure is admitted behind the plungers, and the apparatus driven through the main. At Lancaster, after twice scraping the discharge was increased 56| per cent., at Oswestry 54 ^ per cent. The increased discharge is due to the diminution of the friction of the pipe by re moving the roughnesses due to oxidation. 75. Reduction of a long Pipe of Varying Diameter to an Equiva lent Pipe of Uniform Diameter. Diqmit s Equation. Water mains for the supply of towns often consist of a series of lengths, the dia meter being the same for each length, but differing from length to length. In approximate calculations of the head lost in such mains, it is generally accurate enough to neglect the smaller losses of head and to have regard to the pipe friction only, and then the calcula tions may be facilitated by reducing the main to a main of uniform diameter, in which there would be the same loss of head. Such a uniform main will be termed an equivalent main. 7 -7 - la l/i w B Fig. 84. In fig. 84, let A be the main of variable diameter, and B the equivalent uniform main. In the given main of variable diameter A, let /! , I. 2 . . be the lengths^ d l . d., . . . . the diameters, i , v. 2 . . . . the velocities, ? , >. 2 . . . . the slopes, which gives the length of the equi valent uniform main which would have the same total loss of head for any given discharge. 76. Case of a Pipe of Uniform Diameter with Discharge diminishing uniformly along its Length. In the case of a branch main the water is delivered at nearly equal distances to service pipes along the route. Such a main approximates to the case of a main of uniform diameter, with a discharge at each point diminished by an amount proportional to the distance from the origin. Let AB (fig. 85) be a main of diameter d and length L ; let Q cubic feet per second enter at A, and let q cubic feet be delivered to service pipes per foot of its length. Then at any point C, I feet from A, the discharge is Q = Qo ~ ?^ Consider a short length dl at c. The loss of head in that length is d ~ f 7 Hence the whole head lost in the length AB will be h = -~- 5 I (Q - ql) z dl 7T a _/o _64CL or, putting P = qlj, the total discharge through the service pipes between A and B, a i f-i i .... (2). II -, 11 II I/ ->, 1 7 Fig. 85. The discharge at the end B of the pipe is -" li Qx-Qo-P. If Q is put for the dis charge of the pipe when the flow into the service pipes is stopped, under the same head h, it may be shown that Qi = Q 55 P nearly If the pipe is so long that all the water passing into service pipes en route, (3), 77. Other Losses of Head in Pipes. Most of the losses of head in
pipes, other than that due to surface friction against the pipe, are due
