THE HYDRAULIC GRADIENT.

If the pipe is required to deliver Q ft3/sec, then

481

giving the requisite diameter d for a given delivery Q and hydraulic gradient ✪ in water works; for instance, between a reservoir and a cistern at a lower level.

The slight deviations of a main from a straight level line do not sensibly affect the results of this formula, unless the pipe rises above the hydraulic gradient, when it acts in the manner of a siphon (§ 194).

386. The Resistance of Ships.

The resistance to the motion of a vessel through the water is initially zero, but the resistance mounts up as the velocity increases.

Taking the knot as a speed of 100 ft a minute, one H.P. is equivalent to 330 knot-pounds, or 33÷224 knot-tons; so that if a steamer of W tons displacement is propelled at a speed of V knots, it experiences a resistance the same as that of a smooth gradient of one in 224 WV÷33 H.P.; in the Paris and New York, for instance, if W=10000, V=20, and H.P.=20000, the gradient is about one in 68.

Although no theory is in existence which will enable us to predict with certainty the resistance at a given speed of a vessel of given design, still the experiments of Froude enable us to assign this resistance from the measured resistance of a model of the vessel run at a correspondingly reduced speed.

According to Froude's Law, "The resistance of similar vessels at speeds as the square root of the length, or as the sixth root of the displacement, is as the displacement or as the cube of the length."

482

THE RESISTANCE OF SHIPS.

Then if L, n2L are the lengths, and D, noD the displacements of a vessel and its model, the resistances at speeds V, nV will be R, n°R; and therefore the HP.'s will be as 1 to n.

If the resistance is supposed to be due to skin friction, and this again is supposed to be proportional to the wetted surface, or as 1 to nt, then the remaining factors of the resistance are as 1 to n2, or proportional to the square of the velocity, as in fluid friction.

Thus, for instance, we may take n=01; and if a vessel is designed to have a length L and a speed of V knots, a reduced model of 100th the length is run at onetenth the speed, and the resistance r pounds is measured; then Froude's Law asserts that the full-sized vessel will experience a resistance 10 pounds at V knots, and the effective H.P. required will be 10% V/330.

The coal required per hour is proportional to the H.P. or to n', but the coal per mile is as no or as the displacement; so that over the same length of voyage the coal endurance is the same.

In popular language, to increase the speed one per cent. over a given voyage, we must increase the length two per cent. or the tonnage and coal capacity 6 per cent., and the horse power, boiler capacity, and daily consumption of fuel by 7 per cent.

Thus taking the Paris and New York as the model for a new design of a steamer to cross the Atlantic, 2800 miles, at a speed of 21 knots, then

n=105, n=134, n=140;

so that the new steamer would have about 13,400 tons displacement and require 28,000 H.P.; this is approximately the case in the Campania and Lucania.

The voyage would take 2800÷21=133 hours; but if 5 hours is deducted for longitude difference on the westward voyage, when running before the sun, the apparent time is 128 hours, so that the apparent speed is raised to 21.82 knots.

Examples.

(1) A bucket of water in a balance discharges 4 lb of water per minute through an orifice in its base at 45° to the vertical, and is kept constantly full by a vertical stream which issues from an orifice 8 ft above the surface with velocity 30 f/s.

Prove that the bucket must be counterpoised by about 0.066 lb more than its weight.

(2) The bucket valve in fig. 80 (p. 362) has a small leak, one-800th of the cross section of the barrel, and the height of the water barometer is taken as 32 ft, the height AO as 16 ft, and the specific volume of the air 800 times that of water.

Prove that the pump will not suck unless the bucket is moved with a velocity greater than

but that afterwards water will be lifted if the velocity is greater than 0.04 f/s.

(3) Prove that a hydraulic engine (fig. 78), in which water under pressure is admitted through small orifices to actuate the piston, will do most work when the speed is 3/3 of the unloaded speed, and the load is of the maximum load, and that the efficiency is then 3.

(4) Discuss the influence of inertia and of fluid friction in the pipe, when the Hydraulic Press (§ 12) is actuated by the Accumulator (§ 15). (Cotterill, Applied Mechanics, § 256.)

(5) Prove that the H.P. of the feed pump of a boiler, which evaporates W lb/min of water at a gauge pressure p lb/in2, must exceed

144 Wp÷33000.

(6) Prove that if the jet of § 371 delivers Q ft3/sec, and the hose is ft long, the pumping H.P. of the fire

(7) If water is scooped up from a trough between the rails into a locomotive tender to a height of h ft, determine the minimum velocity required, and the delivery at a given extra speed, taking the frictional losses as represented by a given fraction of the head.

(8) Show how liquid may be raised through a siphon tube, made to revolve about its longer branch,

which is held vertical; and determine the delivery and the mechanical efficiency for a given angular velocity.

(9) Show how to determine the elements of a cyclone from observations at three points.

What is the direction of rotation in the N. and

S. hemispheres?

CHAPTER XII.

GENERAL EQUATIONS OF EQUILIBRIUM.

387. It was proved in Chapter I., §§ 19, 20, that the surfaces of equal pressure and the free surface of a liquid at rest under gravity are horizontal planes; but this assumes that gravity acts in parallel vertical lines.

When we examine more closely the surface of a large sheet of water like the open sea, we find it uniformly curved, so that the surface is spherical; showing that the lines of force of gravity converge to the centre of the Earth; and Archimedes in his diagrams of floating bodies represents them immersed in a spherical ocean.

If three posts are set up, a mile apart in a straight canal, to the same vertical height out of the water, the visual line joining the two extreme posts will, in the absence of curvature by refraction, cut the middle post 8 ins lower; hence it is inferred that the diameter of the Earth in miles is the number of 8 ins in 1 mile, or 7920.

If miles apart, the visual line cuts at a depth 872 ins; for instance, the Channel tunnel 20 miles long, if made level, would rise in the middle 800 ins from the straight chord; but if made straight, it would have a gradient of about one in 400 at the ends, and water reaching to the ends would have a head of 800 inches in the middle.