Ex. 3. A propeller of 21 feet pitch is altered to 18 feet pitch with the same pressure of steam; if the ship make 9 knots with the first pitch, what should she make now? The square of the rates are in indirect proportion to the pitches, that is: Ex. 1. If the pitch of a propeller is 21 feet and the revolutions per minute 64, and the ship makes 11.29 knots per hour: if the pitch be altered to 23 feet with 76 revolutions per minute, what should now be the rate of the ship, allowing the same per centage of slip ? 21 ft. X 64 rev. 23 ft. X 76 rev. :: 1129 knots : x Ans. 1468 knots. Ex. 2. If the mean pressure was 29 lbs. per square inch, the ship running to miles per hour, then taking the mean pressure to be 32 lbs. per square inch: what is the speed of the ship? Speed. Press. lbs. Press. lbs. Ex. 3. A shaft in a marine engine was making 20 revolutions, and the speed was 8 knots: what will be the speed if the revolutions be increased to 25? The revolutions of the crank vary as the cube of the speed. Let V be the required speed. V3 ... V3 25 X 83 = 28 X 512 = 640, .. VV 640 8.617 knots. = Ex. 4. The revolutions of the crank of a marine engine are 24 per minute, and the speed 10 knots: the revolutions are increased to 30: find the increase of speed. Ans. '77 knot. Ex. 5. A steamer from the Tyne to Flambro' Head, a distance of 72 miles, the counter registers 25671 revolutions, the pitch is 18 feet: how much is this per cent. slip? Ex. 6. The speed of a ship is 8 knots per hour with a consumption of 10 tons per day; but the speed is increased to 9 knots per hour, and the consumption to 13 tons per day: what is the rate of cost per day saved? Ex. 7. Verify the above formula by working at full length under the two conditions on Ex. A ship has a voyage of 2640 knots to go; during 5 days she steamed 228, 236, 240, 250, and 232 knots respectively, up to Friday at 9h A.M. On what day and hour will she arrive at her port, supposing she steams 10 knots per hour the rest of the voyage? TO FIND THE MEAN PRESSURE THROUGHOUT 282. Given the pressure at which steam is admitted into the cylinder, and the portion of the stroke at which it is cut off; it is required to find the mean pressure throughout the stroke. A B We now proceed to calculate the work done by the expansion of steam into a larger volume by virtue of its elasticity. As the steam expands, its elastic force and temperature gradually diminish; in order, therefore, to conduct the calculation, it will be necessary to know beforehand the law according to which the diminution takes place. We shall assume, for the sake of simpliFig. 2. city, that Boyle and Marriotte's law applies to the expansion of steam; it may be thus enunciated:-If a given weight of steam be made to vary its volume without changing its temperature, the elastic force of the steam. will vary in the inverse ratio of the volume it is made to occupy; that is, if its volume E is increased two times, its pressure will be about one-half of what it was at first, and so on. G D H F Thus, let ABCD be a cylinder, EF and GH any two positions of the piston; then the pressure in the position EF is the pressure in the position space EFCD. GH, as the space GHCD is to the 283. In this question four different methods of solution are given. Method III is performed by Methods I and II being those mostly used. using a short table of given multipliers, and is a very short and easy method. Method IV is a solution of the question given by Mr. M'Farlane Gray, the Examiner in Steam. RULE CXXVII.-(METHOD I). This is Simpson's Rule for finding the areas of irregular curvilinear figures. 1°. Divide the cylinder into any number of equal parts. (a)-If the steam be out off at 1,,,, or any measure of 12, divide the cylinder into 12 parts. (b)-If cut off at 5ths, divide it into 10 equal parts. (e)-If cut off at 7ths, divide it into 14 equal parts. (d)-If cut off at 8ths, divide the cylinder into 16 parts. 2°. Divide that part of the stroke through which expansion takes place into inch upon any even number of equal parts, and calculate the pressure per square the piston at each division of the stroke, by Marriotte's law. 3°. Take the sum of the extreme pressures in pounds per square inch, four times the sum of the even pressures, and twice the sum of the odd pressures; then, this sum multiplied by one-third of the distance between the consecutive points at which the pressures are taken, will give the work done expansively per square inch of the area of the piston in one stroke, and one-third this sum is the total pressure during expansion.* 4°. To this add the total pressure before expansion and divide the sum by the number of parts into which the cylinder is divided; the result is the whole work done during a single stroke. NOTE.-The work done before the expansion begins is evidently equal to the pressure per square inch multiplied by the distance described before expansion. The whole work done is equal to the sum of the works done before and after expansion. Calculation of work done, by Simpson's rule. The following method of investigating the question, has the advantage of depending on elementary principles. 1 B In the adjoining figure, let A B represent the volume when the B steam is cut off; then, if A B1 = B1 B2 = B2 B3, &c.; A B2, A Bз, A B4, and A B., will represent the volumes when the steam has expanded into double, treble, &c., its original bulk. Let the perpendicular B1 b, represent the original pressure P, then Bb, B3 b3, B4 b4, and B. b, which are respectively ,,,, of B1 b1, will represent the pressures corresponding to the several volumes. A continuous curve which, in this case, is a hyperbolic, passing through b1, b2, bз, ba, and b, will represent, by its ordinates m m', at any point m of the expansion, the pressure corresponding to the volume A m. A B 3 n', m B At the point m the pressure is represented by m m', the work B done therefore by the steam on one square inch while the load is moved upward through the indefinitely small space m m', will be the product of m n' and half the sum of m n' and n n', or the mean pressure mn; but m n × } (m m' + n n') is equal to the trapezium mn n' m', the whole work done, therefore, will be represented by the sum of all these elementary trapeziums, or the area of the curve B1 B, bo b1. 1 2 A The area of the curve may be found very exactly by Simpson's rule, viz. : In the figure the ordinates are P, † P, † P, † P, and † P, and the interval is h, therefore, Р 5 3 Р Area = {P + 3 + 4 ( 22 + 2 ) + = ? } // Р h multiply by 144 a the number of square inches in the piston, and adding the work done before expansion, W = 2.62 w, a result almost identical with the second method here given. As Simpson's rule requires an even number of intervals, it is necessary when the number of volumes is even, as for instance four, in which case there will be only three intervals, to bisect each interval, insert corresponding ordinates, and multiply by one-sixth of the interval h. The ordinates standing at the middle points of the first, second, and third intervals are P, P, and P, taking account merely of the numerical coefficients, the total coefficient may be found as follows: {1 + 4 (3 + 3 + 3) + 2 (} + }) } × 3 = 1*38. to which adding unity for the work done by expansion, the result 2.38. |