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follows also that, with steam to be used expansively, whose elastic force is 2 J times as great as that of steam used unexpansively, if it be cut off when the piston has moved two-fifteenths of the whole length of stroke, the effective momentum will be the same as that which would be produced by the steam of less elasticity when used unexpansively: while the consumption of steam, and therefore of fuel, in the former case is only onethird (= 1\ x Ts) of the consumption in the latter case. It must be observed, however, that, in order to resist a double expansive force of steam, the machinery ought to have a double strength, and would, consequently, be twice as heavy. In the above investigation no notice is taken of the effects of friction on the movement of the piston; this friction, and the imperfect vacuum in the cylinder, are causes of considerable loss of power in all steam-machinery.

l0. Experience seems to show that these retarding forces may, together, be estimated at about one-fifth of the whole power of the steam; and there is a further diminution, when the steam acts expansively, on account of the loss of heat occasioned by the expansion of a gas; and this, when the steam is allowed to expand to double its original volume, has been estimated at about one-twentieth of the whole power. It follows, as is observed by Messrs. Seaward and Capel," that there may come a time during a stroke when the power of the steam becomes less than the force of resistance against the piston, in which case the piston would stop if it were not for the momentum previously acquired. The same gentlemen observe that there must consequently, in practice, be a limit to the expansive principle; and it is concluded that a cylinder having a 3 feet stroke, in which the steam is cut off at one-third of the range, would be nearly as efficient as a cylinder having a 6 feet stroke in which the steam is cut off at one-sixth, the consumption of fuel being equal. It is recommended that, for marine engines, the expansive force of the steam should not exceed 10 or 12 lbs. per square inch above the pressure of the atmosphere; and Messrs. Seaward and Co. propose that, for engines of great power, the steam should be cut off at one half or three-fifths of the stroke.

'Copy of Letter to the Hon. H. L. Corry, M.P., on the use of High Pressure Steam in the Steam-Vessels of the Royal Navy. l846.

11. Marine engines of the present day are said to be from 20 to 50 per cent, more powerful in giving motion to ships than those of former times; this greater speed, aud the diminished consumption of fuel, are due to the adoption of the wave principle in forming the bows of ships, the improved construction of machinery, and the employment of more elastic steam.

12. The only means of propelling ships by the agency of steam which have as yet been brought to the test of experiment, and which have been generally adopted, are the Paddle-Wheel and the Screw ; but both of these, in their forms, have been variously modified.

13. The reciprocating motion of the piston rods in the two steam cylinders of the engine being made to act, by means of cranks, on the shaft or common axle of the paddle-wheels, causes these to take a revolving motion about that axle; and the reaction of the water against the floats or paddle-boards as they revolve, impels the vessel forward.

14. When the paddle-boards are permanently fixed, as they usually are, in planes passing through the shaft, they necessarily enter the water obliquely; and it is only when any one board is in a vertical position, under the shaft, that the reaction of the water against it is direct. In other positions the boards press against the water in directions oblique to the line of the vessel's motion: on entering the water the boards exert a pressure downwards, while in emerging they lift up a body of water, and both these actions cause violent strains and vibrations in the vessel.

15. The Dip, or the immersion of the lowest paddleboard in the water, should in general be equal to the breadth of the board, so that the upper edge may be a-wash, or on a level with the surface of the water. If the dip should be less than this, part of the engine's power would be ineffective in producing the motion of the ship; if greater, part of that power would he spent in overcoming the greater resistance experienced in alternately depressing and raising the water about the entering and emerging boards.

16. The diameters of paddle-wheels should not exceed four and a half times the length of stroke, for this reason, that if more, the " slip " a of the paddle will be great. With a wheel of such proportion the "slip" would be about 20 per cent. The inner edge of the paddle-board should have as nearly as possible the speed of the ship : the slip will then be at a minimum.

17. The length of a paddle-board should be about, or rather more than ^ the diameter of the wheel. When the diameters of the wheels exceed 4\ times the length of stroke, the engines ordinarily constructed are not capable of driving them effectively, so that the power of the engine is not fully developed. This power should correspond to the velocity assigned to the piston, suppose 200 or 220 feet per minute; and, to be enabled to obtain this with a larger wheel the paddle-board must be narrowed, which would augment the slip, and under adverse circumstances this might become very considerable.

18. These are the proportions for sea-going vessels, and the whole power of the engine should be effective when the vessel is at the mean draught of water, viz. the mean between her extreme light, and load-lines. In river vessels, perhaps, a diameter of wheel equal to about four times the length of stroke would be a good proportion. It is evident that the paddle-boards of sea-going vessels should be more deeply immersed than those of vessels which navigate a river, since at sea, on account of the vessel's pitches, the boards are great part of the time out of water.

19. From the known dimensions of the paddle-wheels in several vessels of war, it appears that the diameters of the wheels vary nearly with the square root of the

■ Loss of power caused by the recession of the water aft ward from the paddle-boards.

horse-power of the engine; and, with a vessel whose engine has a power equal to 200 horses, the diameter of the wheel, between the outer extremities of the paddleboards, is about 20 feet; the lengths of the paddles are rather less than half, and the breadths between one-ninth and one-tenth of the diameter. Hence, if the circumference of a wheel 20 feet in diameter be furnished with 20 paddle-boards 2 feet broad, when the upper edge of the lowest vertical board is a-wash, there will be three boards wholly or partly immersed, one will be nearly entering the surface of the water, and a fifth will have just emerged from it.

20. If a vessel be retained at rest in still water while the wheels revolve, the reaction of the water against a paddle-board will be the greatest when the board is in a vertical position in the water, but this will not always be the case when the vessel is free to move by the rotation of the wheel. In order to explain this subject, let S be the centre of the wheel's rotation, and A B the momentary position of a paddle making, with the vertical line S Z, an angle Z S B represented by 6. Let

V be the velocity of the point C (supposed to be the centre of pressure on A B) in a direction perpendicular to the surface of the paddle A B, and V the velocity of the vessel in the water, in a horizontal direction; then, by the Besolution of Forces or Velocities,

V cos. 6 is that velocity in a direction perpendicular to the surface of the paddle A B; therefore V — V cos. 6 will express the relative velocity of the paddle and vessel in the same direction. But the resistance of a fluid against a body moving in it varies with the square of the velocity ;a therefore (V — V cos. 0)2 may denote the force of resistance, or pressure, against the paddle: this being multiplied by V, the product is the efficient


■ Experiments have shown that this rule is very nearly correct notwithstanding the perturbation of the water by the wheel's rotation.

momentum of that resistance in a direction perpendicular to the surface of the paddle; and consequently (v - v cos. ey v cos. e

is the efficient force by which that resistance impels the vessel forward horizontally: which, for the vertical paddle, where 6=0, becomes

(V - V)' V.

21. But in these expressions it is supposed that the paddles are wholly immersed: this is evidently not the case with the oblique paddles when the upper edge of the lowest vertical paddle is on a level with the surface of the water, for then the immersed part of an oblique paddle is expressed by S B — S A sec. 6; or, r being the radius of the wheel to the outer extremity of a paddle-board, and a the difference between r and the breadth (b) of the paddle, it is expressed by r a sec. 6: consequently the ratio between the efficient resistances against a vertical and an oblique paddle will be as

(V _ v)86 v : (v - v cosv ey v <v cos. e - a).

These expressions being put in numbers according to the data, for different values of 6, it will be found that the first will be less than the second till the part of the paddle's breadth which is out of the water causes a diminution of power which more than compensates for the superiority which is due to the obliquity.

Making the differential of this last expression equal to zero, we may obtain the value of 6 which makes the resistance a maximum. Assuming V = % V, r = 10 feet, a = 8 feet, whence 6=2 feet, the greatest resistance takes place when 6 = 18°; and the force on the vertical paddle is, to that on the oblique paddle in this position, as 10 to 10-865.

The resistance against a vertical paddle being thus proved to be less than the resistance against an oblique

{)addle, in the most effective part of the motion of the atter, it follows that to obtain equal speed for two vessels, one of which is furnished with paddles of the ordinary kind, and the other with such as are kept by

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