Then the head equivalent to AP is AP/w or VAP feet, and the velocity of discharge u will be given by

12=2gV.AP.

AP is measured by a siphon gauge in inches of water.

Let

then

i= the difference of pressure in inches of water,

The mean pressure will be known, say P; then V is given by the formula

Taking now the definite case of air, and taking P to be the ordinary atmospheric pressure, the values are

The volume of gas discharged per second per sq. ft. of effective, i.e. contracted, area of orifice is u c. ft., and its weight is therefore given by

so that for a given value of i the weight discharged de

creases as the temperature rises.

The head producing flow is VAP, or substituting becomes

which increases with the temperature; this head, it must be remembered, is feet of the gas, not feet of water.

Thus, as T increases, the head due to a given difference of pressure increases, so that the velocity and volume of discharge increase; but the density decreases faster, so that the weight discharged is less.

If the flow take place through an orifice, coefficients of resistance and contraction must be allowed, and they may be taken as having the same values as for water. It is of course evident that they are much more liable to variation from small causes. In flow through a pipe, the head wasted in overcoming surface friction is given by

the work on page 472 applying to all cases, and, by the law of resistance on page 471, ƒ has the same value.

The discharge in c. ft. per second of a given pipe will accordingly be the same as on page 483, viz.

taking a mean value for f. The weight discharge must contain T, so cannot be expressed as for water.

EXAMPLES.

I. A circular tank, 20 ft. diameter, is constructed of inch iron plates. Find the greatest depth of water in it, the stress not being more than 4000 lbs. per sq. inch of the solid metal.

Ans. If h be the depth, the tank at the bottom is exposed to an internal bursting pressure wh lbs. per sq. ft. Whence h=38.4 ft.

2. The tank above is 120 ft. above the ground. Find the necessary thickness for a 1 inch copper service pipe on the ground level. Allowing 2000 lbs. per sq. inch. Ans. inch.

3. The discharge from a 2 inch circular orifice in the side of a tank 8 feet below the water level just clears the edge of a 100 gallon tank, distant I feet horizontally and 4 feet vertically; also the tank is half filled in 26 seconds. Find the velocity of discharge and the coefficients of velocity and contraction. Ans. 22 f.s. ; .97; .623.

4. Calculate the time required to sink an iron tank 30 feet long, 20 feet broad, and 9 feet deep, the water entering through an orifice 3 ins. diameter in the thin bottom, supposing the tank when empty to float with 5 feet out of water.

Ans. 1 hr. nearly. 5. Find the result of the preceding when a pipe 4 ins. long is fitted square to the orifice inside the tank.

Ans. 1 hr. 18 min.

6. The barometer stands at 30 ins., and the vacuum gauge on a condenser shows 26 ins. of vacuum. The injection orifice is 10 ins. diameter, 8 feet below the sea-level, and the pipe connecting it to the condenser is 5 feet long. Find the quantity of water entering per second. 4ƒ=.021. Ans. 6.55 c. ft.

7. A cylindrical boiler is 12 feet diameter, and the water level is at the diameter from the bottom; the steam pressure is 120 lbs. by gauge. Find the velocity with which water would flow out-Ist, through a hole in the bottom into the stokehold; 2d, through a 6 inch pipe 8 feet long into the sea, the bottom of the boiler being 15 feet below the sea-level. Ans. 130 f.s. ; 92.3 f.s. 8. A 4 inch pipe, running full, delivers 120 gallons of water per minute. Find the hydraulic gradient or virtual slope. Ans.

9. Water issues from the nozzle of a fire hydrant I inch diameter with a velocity sufficient to project the jet to a height of 100 feet. Determine the pressure in the hose near the nozzle, the internal diameter being 3 inches. Neglect the effect of friction.

Ans. 43 lbs. per sq. inch above atmospheric.

10. Find the velocity of flow of water in a rectangular canal

30 feet wide by 5 deep, sloping 18 inches per mile.

of friction .012.

Coefficient

Ans. The head wasted in friction per mile is 1 ft., and the hydraulic mean depth is 3 ft., whence V=2.4 f.s.

11. A pipe 5 ins. diameter delivers a certain quantity of water per minute with a loss of head of 4 ft. Determine the loss of head if the same quantity were delivered through a pipe 4 ins. diameter, assuming the same coefficient of friction.

Ans. 12.2 ft.

12. The diameter of a screw propeller is 15 feet, pitch 18 feet; neglecting slip, find the horse power wasted in overcoming the friction of I sq. ft. of blade at the tip, at 100 revolutions. Coefficient .005. Ans. 10.8.

13. A 1 inch circular hole in the side of a tank is fitted with an expanding nozzle 2 ins. diameter at its open end. Find the greatest depth of water over the hole for which steady flow is possible, neglecting all friction or contraction.

Ans. 2 ft. 4 ins.

14. A siphon 4 ins. diameter, with its end I foot below the water level in the source, discharges water on a level 6 feet lower. The total length is 80 feet, and its highest point is 36 feet from the entrance end. Find the discharge, and the greatest height possible for continuous flow.

Ans. 240 galls. per min.; 30.8 ft. above water level.

15. A 4 inch pipe delivers 100 gallons per minute into a 6 inch pipe, the axis of the two lying in one horizontal line. The pressure in the 4 inch pipe is atmospheric. Find the waste of head at the entrance, and the pressure in the larger pipe.

Ans. Waste of head 3 ft. 6 ins. Had no head been wasted the pressure would be 18.6 lbs. per sq. inch; the waste of head is equivalent to a loss of pressure 1 lbs., hence the pressure is 17.1 lbs. per sq. inch.

16. Obtain the second result of question 7, when a cock in the pipe is half closed, and allowing for two ordinary bends.

Ans. 23 f.s.

17. 1000 c. ft. of water per minute are pumped through the surface condenser of a marine engine, and discharged into the sea through an orifice 27 ins. diameter. The I.H.P. of the pumping engine is 25. Assuming the mechanical efficiency of the engine and pump combined to be .5, estimate the coefficient of resistance referred to the velocity of discharge.

Ans. Work done on water per minute equals 412,500 foot-lbs., which would lift the 1000 c. ft. through 6.6 ft.

The

total head then to produce V and overcome friction is 6.6 ft., whence F=21.5.

18. The difference of pressure between the two ends of a pipe 6 ft. long, 6 ins. diameter, is 3 ins. of water. Find the speed with which air at atmospheric pressure would flow through-Ist, at 60 F.; 2d, at 600° F. Ans. 107 f.s. ; 140 f.s.

19. The air pressure in a stokehold is 2 ins. of water, find the quantity of air which would be discharged per minute through a hole 2 ins. diameter in the casing. Temperature 90° F.

Ans. 53 lbs.

NOTE. To avoid misapprehension, it may be added that cross motions, such as are shown in Fig. 348, page 465, and referred to on page 469 and elsewhere, would not be possible in the absence of friction and discontinuity. For an explanation of the way in which energy is dissipated in fluids by the formation of eddies which are subsequently extinguished by fluid friction, advanced students are referred to chapter xx. of the larger treatise.