HYDRAULICS.] Expanding the term in brackets, (, J_V = A V l-2 ( tj 60t Neglecting the terms after the second, and 489 fall of the two branch mains. Then, according to Dupuit, the best position for the intersection M is that for which or, if the consumption of water occurs uniformly throughout the length of the branch mains, then -,/^vf + 19 for new pipes = 252 for incrusted pipes. . (9a); _ 9.8 79. Arrangement of a Pipe Network for Town s Supply. Exclud ing the service pipes which directly supply the houses, the smallest branch water mains are made 3 to 4 inches in diameter. For the smallest districts supplied, these are sufficient or more than sufficient to convey the neces sary supply, and in that case the only question arising is, to lay them out so that their total length is as small as possible. tl S- y - Thus if there are two places of consumption A and B there is choice of any of the three arrangements shown in fig 95, the prin cipal main being shown by the dark line, and the branch mains by the thinner ones. If, however, the supply through the branch mains requires pipes of more than the minimum diameter, then the condition to be fulfilled is that the sum of the products of the lengths and dia meters should be a minimum, because the cost of the mains when laid in place is very approximate- n Ll/ l M ly proportional to their length and diameter. For a main de livering water to branch mains on each side, the best position is that which makes the branch mains of equal diameter. Suppose water is to be supplied from an inter mediate main along branch mains to a and b (fig. 96). Let Q a , Q& be the quantities to be delivered at a and b, and h , f> the virtual 4 y / Iii this way various points MN may be determined giving the posi tion AMN for the main, and afterwards the nearest convenient position ABO may be fixed. In determining the consumption of water Q for any given locality, the mode of supply must be taken into account. On the inter mittent system, water is supplied for a period of t seconds daily. Then the discharge per second is ^ d , where N is the number of V inhabitants, and Qj the daily supply to each in cubic feet. With a constant supply the rate of flow is variable at different periods of the day, and the maximum rate of flow may be taken at 2 1 times the mean rate. Hence in this case the discharge to be provided for in the mains is i x<i . The daily supply to a district NQj is sometimes taken proportional to the area supplied, sometimes to the length of house frontage in the district. Determination of the Diameters of Different Parts of a Water Main. When the plan of the arrangement of mains is determined upon, and the supply to each locality and the pressure required is ascer tained, it remains to determine the diameters of the pipes. Let fig. 97 show an elevation of a main ABCD . . ., R being the reservoir from which the supply is derived. Let NN be the datum line of the levelling operations, and H a , H& . . . the heights of the main above the datum line, H r being the height of the water surface in the reservoir from the same datum. Set up next heights A Aj, BB 1; . . . representing the minimum pressure height necessary for the adequate supply of each locality. Then AjBjCjDj . . . is a line which should form a lower limit to the line of virtual slope. Then if heights tya, fit, f)<:. . are taken representing the actual losses of head in each length l n , 7&, l c ... of the main, A,,B C will be the line of virtual slope, and it will be obvious at what points such as D and E , the pressure is deficient, and a different choice of diameter of main is required. For any point z in the length of the main, we have Pressure height = H r - IT,- (ha + fy, + k). Where no other circumstance limits the loss of head to be assigned to a given length of main, a consideration of the safety of the main from fracture by hydraulic shock leads to a limitation of the velocity of flow. Generally the velocity in water mains lies between 1 and 44 feet per second. Occasionally the velocity in pipes reaches 10 feet per second, and in hydraulic machinery work ing under enormous pressures even 20 feet per second. Usually the velocity diminishes along the main as the discharge diminishes, so as to reduce somewhat the total loss of head which is liable to render the pressure insufficient at the end of the main. Mr Fanning gives the following velocities as suitable in pipes for towns supply : Diameter in inches 4 8 12 18 24 30 36 Velocity in feet per sec. ... 2 5 3 3 5 4 5 5 3 6 2 7 80. Branched Pipe connecting Reservoirs at Different Levels. Let A, B, C (fig. 98) be three reservoirs connected by the arrangement of pipes shown,/,, d lt Q,, v l ; Z 2 , d 2 ,,Q. 2 , v. 2 ; 1 3 , d 3 , Q 3 , v s being the length, diameter, discharge, and velocity in the three portions of the main pipe. Suppose the dimensions and positions of the pipes known and the discharges required. If a pressure column is introduced at X, the water will rise to a height XR, measuring the pressure at X, and aR, Eb, Re will be the lines of virtual slope. If the free surface level at R is above b, the reservoir A supplies B and C, and if R is below b, A and B supply C. Consequently there are three cases : I. R above b ; Q, = Q 2 + Q 3 . II. R level with b ; Q^Qa ; Q 2 = 0. III. R below b; Q 1 + Q 2 = Q 3 . To determine which case has to be dealt with in the given condi tions, suppose the pipe from X to B closed by a sluice. Then there is a simple main, and the height of free surface/;. at X can be deter mined. For this condition
XII. 62
