will first be tried for the main A. On referring to the hydraulic tables for cast-iron pipes, it is found that, for a diameter of h

24 in. and a discharge of 19 cu. ft. per sec., the value of s, or I'

is .0052; and, since = 10,000, this gives h=10,000.0052 = 52 ft. as the required head between R and O1. As this is greater

than the actual difference in elevation (600-556) between R and O1, the assumed diameter is too small. Trying a 30-in.

pipe, the value of

h

is found to be .0017; therefore, h=10,000

X.001717 ft. This makes the piezometric elevation at Oi 600-17583 ft. As this is greater than 556, the elevation of O1, and also greater than 550, the elevation of B, the 30-in. pipe may be used for the main A. The heads for pipes B and C are, respectively, 583-550-33 and 583-350=233.

h

The corresponding values of are 33÷6,000=.0055, and 233÷4,000= .0583. Knowing these values and the discharges, the diameters can be taken from the table. They are 14 in. for pipe B and 8 in. for pipe C.

In carrying the main to the next branch point 02, the possibilities of choice of size are greater. But since the point H, 11,000 ft. away, is at an elevation of 400, it is desirable to reduce the head as little as may be, and it will be assumed that an effective head of 50 ft. will give necessary pressures without making the pipes too large. The effective head in J being 50

ft. in 2,000, the value of

h

is 50÷2,000= .025; and from the

table, the pipe necessary to carry 12 cu. ft. per sec. with this

value of

h

is found to be between 14 and 16 in. Using the 14-in.

pipe, the value of

h

is .033; h=2,000X.033=66 ft., and,

·

therefore, the piezometric elevation at O2 is 583-66-517 ft.

h

Proceeding as for the branches B and C, the 'value of for E is found to be .0355, which, by the table, requires an h

8-in. pipe; for D,

a 10-in. pipe.

.0195, which, by the table, requires

Still bearing in mind the elevation of 400 at H, an effective head of 50 ft. will be assumed between O2 and Oз, so that the piezometric elevation at the junction Os will be 317-50

= 467. The pipe K, then, will have a value of

h

of 50÷3,000

= .017; and it is found by the table, that for a delivery of 6.5 cu. ft. per sec., a 14-in. pipe is a little too large; it may,

however, be used. The table gives, for that pipe,=.012, and therefore, h=3,000X.012= 36 ft. The piezometric elevation at the junction Os is, then, 517-36-481. Proceeding as before, it is found that each of the branches F and G requires an 8-in. pipe.

Assuming an effective head of 30 ft. for L, the value of

is 30÷2,000.015, and the pipe L is found to be between an 8- and a 10-in. pipe. For the 10-in. pipe, and the delivery

h Z

of 2 cu. ft. per sec., the value of is .0057; therefore, h=.0057

X2,000 11.4 and the piezometric elevation at O is 481.0 -11.4-469.6. The branches I and H are found to require diameters of 8 and 6 in., respectively.

HYDRAULIC GRADE LINE

The hydraulic grade line, or hydraulic gradient, is a line drawn through a series of points to which water would rise in piezometer tubes attached to a pipe through which water flows. With a straight smooth pipe of uniform cross-section, the hydraulic grade line is a straight line extending from the reservoir to the end of the pipe.

In the accompanying illustration is shown a horizontal pipe leading from a reservoir to a stop-valve S. When the valve is. open so that water from the pipe discharges freely into the atmosphere, the hydraulic grade line is the line adfg. The distance of the point a below the surface of the water in the reservoir represents the head absorbed in overcoming the resistances of entrance to the pipe, and in producing the velocity with which the water flows. In the same way, the difference in the height to which the water rises in any two piezometer tubes represents the head absorbed in overcoming the resistance to flow in the pipe between the points at which the tubes are inserted.

The flow of water through the pipe P would be the same whether the pipe were horizontal, as shown in the illustration, or whether it were laid along the grade line adfg. The flow

would also be the same if the reservoir were deepened and the pipe laid along the line a'd'f'. The pressures in the pipe, however, would vary greatly with the different positions. If the pipe were laid along the line adfg, there would be little or no pressure in any part of it. In the horizontal position, however,

and still more in the position a'd'f', there would be pressure at all points, the pressure for any point in the pipe being equivalent to the head represented by the vertical distance from that point to the hydraulic grade line.

Position of Hydraulic Grade Line.-In laying a line of pipe to connect two points lying at different levels, it is of the utmost importance to ascertain the position of the hydraulic grade line. In order that the pipe may flow full, no part of it should rise above the hydraulic grade line.

The Siphon. The part of a pipe that rises above the hydraulic gradient is called a siphon. If the siphon is kept filled, the flow through it will take place in accordance with the laws given for pipes laid below the hydraulic gradient, and the same formulas apply.

The total head producing the flow in a siphon is the vertical distance from the discharge end of the pipe to the level of the water in the reservoir, but the pressure in all parts of

H

the pipe that rise above the line will be less than the atmospheric pressure. Air always tends to collect in the highest point of a siphon, and means must be provided for its removal, in order to keep up the flow. This is effected by means of an air pump or air valve. Such means of removing the air should be provided for whenever circumstances make it unavoidable to place part of a pipe above the hydraulic gradient.

CAST-IRON PIPES

The thickness of a cast-iron pipe may be computed by the

in which is the thickness of pipe, in inches; p, the static pressure, due to the head above the pipe, in pounds per square inch; d, the diameter of pipe, in inches; and p', the allowance for water hammer (shocks caused by opening of valves). The following are values of p' for different diameters:

EXAMPLE.-Determine the thickness of a cast-iron pipe 14 in. in diameter to withstand a pressure of 130 lb. per sq. in.

SOLUTION.-Here, d=14 and p=130. The value of p corresponding to a diameter of 14 in. is a mean between the values corresponding to the diameters 12 and 16, or 105. Substituting these values in the formula,

Weight of a Cast-Iron Pipe Line.-To ascertain by a rapid approximation the weight, in tons (2,000 lb.), of a cast-iron pipe line, the following formula may be used:

T=28mt (d+t)