Equation of Pipes.-It is frequently desired to know what number of pipes of a given size are equal in carrying capacity to one nipe of a larger size. At the same velocity of flow the volume delivered by two pipes of different sizes is proportional to the squares of their diameters; thus, one 4-inch pipe will deliver the same volume as four 2-inch pipes. With the same head, however, the velocity is less in the smaller pipe, and the volume delivered varies about as the square root of the fifth power (i.e., as the 2.5 power). The following table has been calculated on this basis. The figures opposite the intersection of any two sizes is the number of the smaller-sized pipes required to equal one of the larger. Thus, one 4-inch pipe is equal to 5.7 2-inch pipes. 11.7 5.7 3.2 2.1 1.4 1 9 243 43. 15.6 7.6 4.3 2.8 1.9 1.3 1 10 316 55.9 20.3 9.9 5.7 3.6 2.4 1.7 1.3 1 11 401 70.925.712.5 7.2 4.6 3.1 2.2 1.7 1.3 12 499 88.232 15.6 8.9 5.7 3.8 2.8 2.1 1.6 1 13 609 108 39.1 19 10.9 7.1 4.7 3.4 2.5 1.9 1.2 14 733 130 47 22.9 13.1 8.3 5.7 4.1 3.0 2.3 1.5 1 15 871 154 55.9 27.2 15.6 9.9 6.7 4.8 3.6 2.8 1.7 1.2 181 65.732 18.311.7 7.9 5.7 4.2 3.2 2.1 1.4 1 211 76.4 37.2 21.3 13.5 9.2 6.6 4.9 3.8 2.4 1.6 1.2 243 88.2 43 24.6 15.6 10.6 7.6 5.7 4.3 2.8 1.9 1.3 1 278 101 49.1 28.1 17.8 12.1 8.7 6.5 5 3.2 2.1 1.5 1.1 316115 55.932 20.3 13.8 9.9 7.4 5.7 3.6 2.4 1.7 1.3 1 401 146 70.9 40.6 25.7 17.5 12.5 9.3 7.2 4.6 3.1 2.2 1.7 1.3 88.250.532 21.8 15.6 11.6 8.9 5.7 3.8 2.8 2.1 1.6 1 108 61.739.126.6 19. 14.2 10.9 7.1 4.7 3.4 2.5 1.9 1.2 74.247 32 22.9 17.1 13.1 8.3 5.7 4.1 3 2.3 1.5 871 316 154 88.2 55.938 27.2 20.3 15.6 9.9 6.7 4.8 3.8 2.8 1.7 499 243 130 88.2 60 43 32 24.6 15.6 10.6 7.6 5.7 4.3 2.8 733 357 205 130 88.2 63.2 47 36.2 19 499 286 181 123 88.2 62.7 50.5 32 670 383 243 165 118 88.2 67.843 17 18 19 20 22 24 26 28 30 36 42 48 54 60 499 181 609 221 733 266 130 15.6 11.2 8.3 6.4 4.1 21.8 15.6 11.6 8.9 5.7 871 499 316 215 154 115 88.2 55.9 38 27.2 20.3 15.6 9.9 Measurement of the Velocity of Air in Pipes by an Anemometer.-Tests were made by B. Donkin, Jr. (Inst. Civil Engrs. 1892), to compare the velocity of air in pipes from 8 in, to 24 in. diam., as shown by an anemometer 234 in. diam. with the true velocity as measured by the time of descent of a gas-holder holding 1622 cubic feet. A table of the results with discussion is given in Eng'g News, Dec. 22, 1892. In pipes from 8 in. to 20 in. diam. with air velocities of from 140 to 690 feet per minute the anemometer showed errors varying from 14.5% fast to 10% slow. With a 24-inch pipe and a velocity of 73 ft. per minute, the anemometer gave from 44 to 63 feet, or from 13.6 to 39.6% slow. The practical conclusion drawn from these experiments is that anemometers for the measurement of velocities of air in pipes of these diameters should be used with great caution. The percentage of error is not constant, and varies considerably with the diameter of the pipes and the speeds of air. The use of a baffle, consisting of a perforated plate, which tended to equalize the velocity in the centre and at the sides in some cases diminished the error. The impossibility of measuring the true quantity of air by an anemometer held stationary in one position is shown by the following figures, given by Wm. Daniel (Proc. Inst. M. E., 1875), of the velocities of air found at different points in the cross-sections of two different airways in a mine, DIFFERENCES OF ANEMOMETER READINGS IN AIRWAYS. Force of the Wind.-Smeaton in 1759 published a table of the velocity and pressure of wind, as follows: VELOCITY AND FORCE OF WIND, IN POUNDS PER SQUARE INCH. The pressures per square foot in the above table correspond to the formula P=0.005, in which V is the velocity in miles per hour. Eng'g News, Feb. 9, 1893, says that the formula was never well established, and has floated chiefly on Smeaton's name and for lack of a better. It was put forward only for surfaces for use in windmill practice. The trend of modern evidence is that it is approximately correct only for such surfaces, and that for large solid bodies it often gives greatly too large results. Observations by others are thus compared with Smeaton's formula: Old Smeaton formula... As determined by Prof. Martin... 66 Whipple and Dines... .P = .005Va dQv g At 60 miles per hour these formulas give for the pressure per square foot, 18, 14.4 and 10.44 lbs., respectively, the pressure varying by all of them as the square of the velocity. Lieut. Crosby's experiments (Eng'g, June 13, 1890), claiming to prove that PfV instead of P = fV, are discredited. A. R. Wolff (The Windmill as a Prime Mover, p. 9) gives as the theoretical pressure per sq. ft. of surface, P= in which d = density of air in pounds .018743(p+P). per cu. ft. = ;p being the barometric pressure per square t foot at any level, and temperature of 32° F., t any absolute temperature, Q volume of air carried along per square foot in one second, v = velocity dv of the wind in feet per sec., y = 32.16. Since Q : vcu. ft. per sec., P= Multiplying this by a coefficient 0.98 found by experiment, and substituting =2116.5 lbs. per sq. ft. or average atmospheric pressure at the sea-level, 36.8929 P= an expression in which the pressure is shown to vary tx 32.16 v2 -0.18743 with the temperature; and he gives a table showing the relation between velocity and pressure for temperatures from 0° to 100° F., and velocities from 1 to 80 miles per hour. For a temperature of 45° F. the pressures agree with those in Smeaton's table, for 0° F. they are about 10 per cent greater, and for 100 10 per cent less. Prof. H. Allen Hazen, Eng'g News, July 5, 1890, says that experiments with whirling arms, by exposing plates to direct wind, and on locomotives with velocities running up to 40 miles per hour, have invariably shown the resistance to vary with V. In the formula P = .005SV2, in which P = pressure in pounds, S = surface in square feet, V velocity in miles per hour, the doubtful question is that regarding the accuracy of the first two factors in the second member of this equation. The first factor has been variously determined from .003 to .005 [it has been determined as low as .0014.-Ed. Eng'g News]. The second factor has been found in some experiments with very short whirling arms and low velocities to vary with the perimeter of the plate, but this entirely disappears with longer arms or straight line motion, and the only question now to be determined is the value of the coefficient. Perhaps some of the best experiments for determining this value were tried in France in 1886 by carrying flat boards on trains. The resulting formula in this case was, for 44.5 miles per hour, p = .00535SV2. Mr. Crosby's whirling experiments were made with an arm 5.5 ft. long. It is certain that most serious effects from centrifugal action would be set up by using such a short arm, and nothing satisfactory can be learned with arms less than 20 or 30 ft. long at velocities above 5 miles per hour. Prof. Kernot, of Melbourne (Engineering Record, Feb. 20, 1894), states that experiments at the Forth Bridge showed that the average pressure on surfaces as large as railway carriages, houses, or bridges never exceeded two thirds of that upon small surfaces of one or two square feet, such as have been used at observatories, and also that an inertia effect, which is frequently overlooked, may cause some forms of anemometer to give false results enormously exceeding the correct indication. Experiments of Mr. O. T. Crosby showed that the pressure varied directly as the velocity, whereas all the early investigators, from the time of Smeaton onwards, inade it vary as the square of the velocity. Experiments made by Prof. Kernot at speeds varying from 2 to 15 miles per hour agreed with the earlier authorities, and tended to negative Crosby's results. The pressure upon one side of a cube, or of a block proportioned like an ordinary carriage, was found to be .9 of that upon a thin plate of the same area. The same result was obtained for a square tower. A square pyramid, whose height was three times its base, experienced .8 of the pressure upon a thin plate equal to one of its sides, but if an angle was turned to the wind the pressure was increased by fully 20%. A bridge consisting of two plate-girders connected by a deck at the top was found to experience 9 of the pressure on a thin plate equal in size to one girder, when the distance between the girders was equal to their depth, and this was increased by one fifth when the distance between the girders was double the depth. A lattice-work in which the area of the openings was 55% of the whole area experienced a pressure of 80% of that upon a plate of the same area. The pressure upon cylinders and cones was proved to be equal to half that upon the diametral planes, and that upon an octagonal prism to be 20% greater than upon the circumscribing cylinder. A sphere was subject to a pressure of .36 of that upon a thin circular plate of equal diameter. A hemispherical cup gave the same result as the sphere; when its concavity was turned to the wind the pressure was 1.15 of that on a flat plate of equal diameter. When a plane surface parallel to the direction of the wind was brought nearly into contact with a cylinder or sphere, the pressure on the latter bodies was augmented by about 20%, owing to the lateral escape of the air being checked. Thus it is possible for the security of a tower or chimney to be impaired by the erection of a building nearly touching it on one side. Pressures of Wind Registered in Storms.-Mr. Frizell has examined the published records of Greenwich Observatory from 1849 to 1869, and reports that the highest pressure of wind he finds recorded is 41 lbs. per sq. ft., and there are numerous instances in which it was between 30 and 40 lbs. per sq. ft. Prof. Henry says that on Mount Washington, N. H., a velocity of 150 miles per hour has been observed, and at New York City 60 miles an hour, and that the highest winds observed in 1870 were of 72 and 63 miles per hour, respectively. Lieut. Dunwoody, U. S. A., says, in substance, that the New England coast is exposed to storms which produce a pressure of 50 lbs. per sq. ft. Engineering News, Aug. 20, 1880. WINDMILLS. = Power and Efficiency of Windmills.-Rankine, S. E., p. 215, gives the following: Let Q = volume of air which acts on the sail, or part of a sail, in cubic feet per second, v = velocity of the wind in feet per second, s sectional area of the cylinder, or annular cylinder of wind, through which the sail, or part of the sail, sweeps in one revolution, c = a coefficient to be found by experience; then Q cvs. Rankine, from experimental data given by Smeaton, and taking c to include an allowance for friction, gives for a wheel with four sails, proportioned in the best manner, c = 0.75. Let A weather angle of the sail at any distance from the axis, i.e., the angle the portion of the sail considered makes with its plane of revolution. This angle gradually diminishes from the inner end of the sail to the tip; u = the velocity of the same portion of the sail, and E the effi ciency. The efficiency is the ratio of the useful work performed to whole energy of the stream of wind acting on the surface s of the wheel, which Dsv3 energy is D being the weight of a cubic foot of air. Rankine's formula 2g in which c = 0.75 and ƒ is a coefficient of friction found from Smeaton's data = 0.016. Rankine gives the following from Smeaton's data: Rankine gives the following as the best values for the angle of weather at different distances from the axis: Distance in sixths of total radius... 1 66 ............. 2 3 4 5 6 18. 19° 18° 16° 121⁄2° 7° But Wolff (p. 125) shows that Smeaton did not term these the best angles, but simply says they answer as well as any," possibly any that were in existence in his time. Wolff says that they cannot in the nature of things be the most desirable angles." Mathematical considerations, he says, conclusively show that the angle of impulse depends on the relative velocity of each point of the sail and the wind, the angle growing larger as the ratio becomes greater. Smeaton's angles do not fulfil this condition. Wolff devel ops a theoretical formula for the best angle of weather, and from it calculates a table for different relative velocities of the blades (at a distance of one seventh of the total length from the centre of the shaft) and the wind, from which the following is condensed: Distance from the axis of the wheel in sevenths of radius. The effective power of a windmill, as Smeaton ascertained by experiment, varies as s, the sectional area of the acting stream of wind; that is, for similar wheels, as the squares of the radii. The value 0.75, assigned to the multiplier c in the formula Q = cvs, is founded on the fact, ascertained by Smeaton, that the effective power of a windmill with sails of the best form, and about 151⁄2 ft. radius, with a breeze of 13 ft. per second, is about 1 horse-power. In the computations founded on that fact, the mean angle of weather is made 13°. The efficiency of this wheel, according to the formula and table given, is 0.29, at its best speed, when the tips of the sails move at a velocity of 2.6 times that of the wind. Merivale (Notes and Formulæ for Mining Students), using Smeaton's coefficient of efficiency, 0.29, gives the following: U W units of work in foot-lbs. per sec.; weight, in pounds, of the cylinder of wind passing the sails each second, the diameter of the cylinder being equal to the diameter of the sails; V= velocity of wind in feet per second; effective horse-power; H.P. WV2 0.29 WV2 J = ; H.P. = 64 64 X 550 A. R. Wolff, in an article in the American Engineer, gives the following (see also his treatise on Windmills): Let c = velocity of wind in feet per second; n = number of revolutions of the windmill per minute; bu, b1, b2, by be the breadth of the sail or blade at distances lo, 11, 12, la, and 1, respectively, from the axis of the shaft; lo : = distance from axis of shaft to beginning of sail or blade proper; 1= distance from axis of shaft to extremity of sail proper; Vo, V1, V2, V3, V = the velocity of the sail in feet per second at dis tances lo, 11, 12, 1, respectively, from the axis of the shaft; ao, A1, A2, A3, аx the angles of impulse for maximum effect at distances lo, 11, 12, 13, 1 respectively from the axis of the shaft; a = the angle of impulse when the sails or blocks are plane surfaces, so that there is but one angle to be considered; N= number of sails or blades of windmill; K = .93. d = density of wind (weight of a cubic foot of air at average tempera ture and barometric pressure where mill is erected); W = weight of wind-wheel in pounds; f coefficient of friction of shaft and bearings; = diameter of bearing of windmill in feet. |