TEETH OF WHEELS. To find the Horses-Power that the Teeth will transmit. Rule. Multiply one fourth of the square of the pitch in inches by the breadth of the teeth in inches; the product is the horsespower that the teeth will transmit when the pitch line passes through 4 feet per second. In quick speeds, or fractional pitches, it may be more convenient to take the following Rule. Multiply the square root of the pitch in inches, by the breadth of the teeth in inches; the product is the horses-power at 16 feet per second. of of the A general rule to ascertain the length of the teeth is to take the pitch for the distance from the root to the pitch line, and pitch for the distance from the pitch line to the top. Hicks' Rule for Calculating Strength of Shafts. Rule.-Multiply the horses-power by the assumed number, (300) and divide the product by the revolutions per minute; the cube root of the quotient will be the diameter required. OF WATER-WHEELS. The properties of water, as a motive power, are gravity and impul sive force, each being rendered peculiarly available for the production of uniform circular motion through the medium of the water-wheel. Water-wheels are necessarily and designedly of various modifications, so as to obtain the greatest amount of mechanical effect, from a known quantity of water flowing at a certain velocity, or from a given height, and generally ranked, by estimation of effect, into first, second, and third-class wheels. 1st class includes overshot wheels, pitch-back wheels, and turbines. 2d class consists of breast wheels, or those which receive the water below the level of the axis. And 3d class is composed of undershot wheels, tub wheels, and flutter wheels. The most modern and best-conducted experiments on each description, as known to the public at present, are those by Poncelot of America, and of Morin in France, the results of which are as follows: Overshot wheels, &c. ; ratio of power to effect, varies fm. 0·60 to 0.80 Breast wheels Undershot wheels, &c. 66 66 66 0.45 to 0.50 0:27 to 0:30 The greatest effect is obtained by an overshot wheel when the diameter of the wheel is so proportioned to the height of the fall, that the water shall flow upon the wheel at a point about 52 degrees distant from the top of the wheel. If r represent the radius of the wheel to the extreme part of the buckets, and h the effective height of the fall, then h=r (1+sin. 374) or h-1 605 r; for the sine 374605. Also 623 h=r. Therefore, when the effective height of the fall is determined, the radius of the wheel is easily obtained. When the effective fall is ths of the whole fall, if h is made the whole fall, r=554 h, or 1·108 h the diameter of the wheel. The effective height of the fall is less than the true height, by as much as is required to give the water a proper velocity before it flows upon the wheel. And modern practice dictates a velocity of about 6 feet per second for the velocity of wheels in general: hence, if the velocity of the wheel be given, divide twice the square of the circumference in feet, per second, by 64 38, and the quotient equal the height of the fall. If the portion of the total descent passed through by the water be given, then the velocity of the circumference should be one-half of that, due to this height. Therefore, multiply the portion of fall, in feet, by 64 38, and the square root of the product equal the water's velocity, in feet, per second. Also, If the area of cross section of the overflow be multiplied by the velocity at the end of the fall, the product equal the quantity, in cubic feet, per second. Observations from Experiments on Overshot Wheels. By R. MALLET, M. In. C. E. From the Transactions of the Institution of Civil Engineers. 1. When the depth of water in the reservoir is invariable, the diameter of the wheel should never exceed the entire height of the fall, less so much as is requisite to generate a proper velocity on entering the buckets. 2. Where the depth of water in the reservoir varies considerably and unavoidably, an advantage may be obtained by applying a larger wheel, dependant upon the extent of fluctuation, and ratio in time, that the water is at its highest or lowest levels during a given prolonged period. If this be a ratio of equality, in time there will be no advantage; and hence, in practice, the cases will be rare when any advantage will be obtained by the use of an overshot wheel greater in diameter than the height of fall, minus the head due to the required velocity of the water reaching the wheel. 3. If the level of the water in the reservoir never fall below the mean depth of the reservoir, when at the highest and lowest, and the average depth be between an eighth and a tenth of the height of the fall, then the average mechanical force of the large wheel will be greater than that of the small one; and it will of course retain its increased advantage at periods of increased depths of the reservoir. 4. That a positive advantage is gained by a wheel revolving in a conduit, varying with the conditions of the wheel and fall of nearly 11 per cent. of the total power. To ascertain the Power of a Water-wheel. Rule.-Multiply the velocity of the wheel, in feet, per minute, by the weight of the water, in lbs., expended on the wheel in the same time; divide the product by the co-efficient of power to effect, and the quotient equal the mechanical effect of the wheel, expressed in horses' power. Or multiply the product of the quantity of water expanded, in cubic feet, per minute, and the velocity of the wheel, in feet, in the same time, by the following decimal equivalents: the product will be the number of horses' power that the wheel is equal to in useful effect. 1st class wheels 001325 { Decimal equivalents 2d class do. 000902 3d class do. 000541 Example-Suppose a stream of water flowing on an overshot wheel at the rate of 95 cubic feet per minute, and the velocity of the wheel's periphery equal 6 feet per second, or 360 feet per minute: required the effect of the wheel in horses' power? NOTE. Where the fall of water do not exceed 4 feet, an undershot wheel ought to be applied; from 4 to 10 feet, a breast wheel; and from 10 feet upwards, an overshot, or pitch-back wheel. To ascertain the Power of a Stream. Rule. Multiply the weight of the water in lbs. discharged in one minute by the height of the fall in feet; divided by 33000, and the quotient is the answer. Example-What power is a stream of water equal to of the following dimensions, viz.: 1 foot deep by 22 inches broad, velocity 350 feet per minute, and fall 60 feet; and what should be the size of the wheel applied to it? 12×22×350×12÷1728 × 62 × 60 feet÷33000=72.9. Ans. Height of fall 60 feet, from which deduct for admission of water, and clearance below, 15 inches, which gives 58.9 feet for the diameter of the wheel. The power of a stream, applied to an overshot wheel, produces effect as 10 to 6.6. Then, as 10 66 729 48 horses' power equal that of an overshot wheel of 60 feet applied to this stream. When the fall exceeds 10 feet, the overshot wheel should be applied. The higher the wheel is in proportion to the whole descent, the greater will be the effect. The effect is as the quantity of water and its perpendicular height multiplied together. The weight of the arch of loaded buckets, in pounds, is found by multiplying of their number, × the number of cubic feet in each, and that product by 40. To ascertain the Power of an Undershot Wheel when the Stream is confined to the Wheel. Rule. Ascertain the weight of the water discharged against the floats of the wheel in one minute by the preceding rules, and divide it by 100000; the quotient is the number of horses' power. NOTE. The 100000 is obtained thus: The power of a stream, applied to an undershot wheel, produces effect as 10 to 33, then 33 10: 33000: 100000. When the opening is above the center of the floats, multiply the weight of the water by the height, as in the rule for an overshot wheel. Example.-What is the power of an undershot wheel, applied to a stream 2 by 80 inches, from a head of 25 feet? √25×65×60=1950 feet velocity of water per minute, and 2×80 =160 inches x 1950 × 12 × 1728=2166 6 cubic feet × 62.5=*135412 lbs. of water discharged in one minute; then 135412÷100000=1·35 horses' power. NOTE. The maximum work is always obtained when the velocity of the wheel is half that of the stream. Let V represent velocity of float boards, and v velocity of (v-V)2 water; then V2 X force of the water, will be the force of the effective stroke. The effect of an undershot wheel to the power expended is, at a medium, one-half that of an overshot wheel. The virtual or effective head being the same, the effect will be very nearly as the quantity of water expended. When the fall is below 4 feet, an undershot wheel should be applied. To find the Power of a Breast Wheel. Rule.-Find the effect of an undershot wheel, the head of water of which is the difference of level between the surface and where it strikes the wheel (breast), and add to it the effect of that of an overshot wheel, the height of the head of which is equal to the difference between where the water strikes the wheel, and the tail water; the sum is the effective power. Example.-What would be the power of a breast wheel applied to a stream 2×80 inches, 14 feet from the surface, the rest of the fall being 11 feet? ✓14×65×60=1458 6 feet velocity of water per minute. And 2× 80 × 1458 × 12÷1728=1620 cubic feet × 62.5=101250 lbs. of water discharged in one minute. Then 101250 100000 1012 horses' power as an undershot. ✓11×65×60-1290 feet velocity of water per minute. And 2×80 × 1290 × 12÷1728=1433 cubic feet × 62.5=89562 lbs of water discharged in one minute. * Equal 160 × 12÷ 1728 × 62·5 × 1950 momentum of water and its velocity. × 11 height of fall÷50000=19 703 horses, which, added to the above, 20 715. Ans. NOTE. When the fall exceeds 10 feet, it may be divided into two, and two breast wheels applied to it When the fall is between 4 and 10 feet, a breast wheel should be applied. The power of a water-wheel ought to be taken off opposite to the point where the water is producing its greatest action upon the wheel. Remarks on Reaction Water-wheels. From the Journal of the Franklin Institute, and other sources. Reaction water-wheels are a very numerous family, of which the well-known hydraulic motor, called Barker's mill, is the parent: those used in various parts of the United States have usually vertical axes of rotation, and curved buckets, or vanes, against which the impulsive force of the water (spouting from within the wheel by adjutages, of which the curved vanes form the sides) acts indirectly, or rather reacts, thus producing (in reference to the affluent water) a backward rotary motion, similar, in character and effect, to the forward rotary motion produced by direct impulse in the case of undershot wheels. In the American Philosophical Transactions for 1793, it is stated that the principles of reaction wheels had been fully investigated analytically in examining the merits of Rumsey's improvements on Barker's mill; and the conclusion come to, after a train of reasoning based upon scientific principles, was, that "action and reaction are equal;" that the undershot wheel is propelled by the action; and Barker's mill by the reaction of the same agent, or momentum: therefore their mechanical effects must be equal. This conclusion no doubt tended to retard any effort at improvement of wheels on that principle for a considerable length of time; for it is only, comparatively speaking, quite recently, that reaction water-wheels, of the form at present in use, have occupied a prominent position before the public. In 1830, Calvin Wing, of the United States, took out a patent for a reaction water-wheel with curved vanes or buckets, the vanes of which lapped over, or rather on to each other, in the ratio of 14 inches, for each inch of the width of the adjutage, or shortest horizontal distance between any two adjacent vanes. In this wheel the water has free entrance to a circular space within, and, spouting out by the openings between the curved vanes, impels the wheel around in a backward direction, by its reaction against the vanes, in issuing with velocity from within the wheel. But this species of wheel, so far, seems not to realize the amount of effect as anticipated; for, according to recent experiments, it appears that, with 788 cubic feet of water, at the rate of one foot per minute, applied on an overshot wheel, will grind and dress one bushel of wheat per hour; whereas to do the same by means of the reaction wheel required 1600. Some of later date, as the turbine of France, by M. Fourneyron, and the recently-patented water-mill, by Whitelaw and Stirrat, Scot land, seem much improved hydraulic motors; for, according to the experiments of M. Morin, and others of high authority, they rank, in effect to power, equal to first-class wheels. |