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The resistance thus determined, being deduced from the work performed in producing eddies, includes in one quantity both the direct adhesive action of the water on the ship's skin and the indirect action, through increase of the pressure at the bow and diminution of the pressure at the stern.
The constant part of the expression deduced by Professor Weisbach, from experiments on the flow of water in pipes, viz., f= .0036, has given a coefficient of friction, corroborated by practice, for surfaces of clean painted iron.
For clean copper sheathing, it appears probable that the coefficient of friction is somewhat smaller, but there are not yet. sufficient experimental data to decide that question exactly. Experimental data are also wanting to determine the precise increase of the coefficient of friction produced by various kinds and degrees of roughness and foulness of the ship's bottom, but it is certain that that increase is sometimes very great.
The preceding value of the coefficient of friction leads to the following very simple rule:-At ten knots, the eddy-resistance of a clean iron ship is one pound avoirdupois per square foot of augmented surface; and it varies, for other speeds, as the square of the speed.
Computation of Augmented Surface.-To compute the exact augmented surface of a vessel of any ordinary shape, would be a problem of impracticable labor and complexity. The method employed, therefore, as an approximation for practical purposes, is to choose in the first instance a figure approximating to the actual figure, but of a kind such that its augmented surface can be calculated by a simple and easy process, and to use that angmented surface instead of the exact augmented surface of the ship; care being taken to ascertain, by comparison with experiments on ships of various sizes and forms, whether the approximation so obtained is sufficiently accurate.
The figure chosen for that purpose is the trochoïd, or rollingwave curve, extending between a pair of crests; for by an easy integration, it is found, that the augmented surface of a trochoïdal riband of a given length in a straight line, and of a given breadth, is equal to the product of that length and breadth, multiplied by the following coefficient of augmentation: 1 + 4 (sine of greatest obliquity)2 + (sine of greatest obliquity)*; the greatest obliquity meaning the greatest angle made by the riband with its straight chord.
In approximating to the augmented surface of a given ship by the aid of that of a trochoïdal riband, the following values are employed :--I. For the length of the riband, the length of the ship on the plane of flotation; II. For the total breadth of the riband, the mean immersed girth, found by measuring, on the body plan, the immersed girths of a series of cross-sections, and taking their mean by Simpson's Rule; III. For the coefficient of augmentation, the mean of the values of that coefficient as deduced from the greatest angles of obliquity of the series of water-lines of the fore-body, as shown on the half-breadth plan. The augmented surface is then computed by multiplying together these three factors.
The Computation of the Probable Resistance (in lbs.), at a given speed, is performed according to the rule already stated, by multiplying the augmented surface by the square of the speed in knots, and dividing by 100 (for clean painted iron ships).
In Computing the Probable Engine-Power required at a given Speed, allowance must be made for the power wasted through slip, through wasteful resistance of the propeller, and through the friction of the engine. The proportion borne by that wasted power to the effective or net power, employed in driving the vessel, of course varies considerably for different ships, propellers, and engines; but in several good examples it has been found to differ little from 0.63; so that as a probable value of the indicated power required, in a well-designed vessel, we may take net power × 1.63.
Now an indicated horse-power is 550 foot-pounds per second, and a knot is 1.688 feet per second; therefore an indicated horse550 power is =326 knot-pounds, nearly; or 326 lbs. gross resist1.688 ance overcome through one nautical mile in an hour. estimate, then, the net or useful work done in propelling the vessel as being equal to the total work of the steam divided by 1.63, we
326 shall have 200 knot-pounds of net work done in propulsion 1.63 for each indicated horse-power. Hence the following rule:Multiply the augmented surface in square feet by the cube of the speed in knots, and divide by 20,000; the quotient will be the probable indicated horse-power.
The divisor in this rule (20,000) expresses the number of square feet of augmented surface, which can be driven at one knot
by one indicated horse-power: it may be called the "coefficient of propulsion."
It is, of course, to be understood that the exact divisor ("or coefficient of propulsion") differs in different vessels, according to the smoothness of the skin, the nature of its material, and the efficiency of the engines and propellers; it being greatest in the most favorable examples.
The value 20,000 may be taken as a probable and safe estimate of the divisor in any proposed vessel designed on good principles. For copper sheathing and smooth pitch, the coefficient of propulsion is certainly greater than 20,000, but in what precise proportion it is at present difficult to decide.
Computation of Probable Speed. When the augmented surface of a ship has been determined, her probable speed, with a given power, is computed as follows:-Multiply the indicated horse-power by 20,000; divide by the augmented surface, and extract the cube root of the quotient for the probable speed in knots.
Example.-Calculation of Probable Speed of H. M. S. Warrior. -Displacement on trial, 8,997 tons; draught of water forward, 25.83 feet; aft, 26.75 feet:
1+(4×.0674) +.0058=1.275, Co-efficient of Augmentation.
Divide by Augmented Surface, 36,979)109,420,000 product.
NOTE.-From Transactions of the Institute of Naval Architects. Vol. v., 1864.