HYDRODYNAMICS. 



553 



Ovi'-t: >l 



\VhccL-. 



wheels, that the diameter of overshot wheels should al- 

 ways be less than the height of the fall of water by which 

 it is to be put in motion, and various ratios have been 

 io brtweVn assigned between the height of the fall and the diaone- 

 tke dUme- ter of the wheel. The Chevalier de Borda has shewn, 

 trs/owr- that overshot wheels will produce a maximum effect 

 hot wheels when their diameter is equal to the greatest height of the 

 tall, but that a slight diminution of the wheel's diameter 

 produces only a very small diminution of the maximum 

 effect. If the height of the fall, for example, is 12 feet, 

 and if the diameter of the whtel is made only 1 1 feet, 

 the effect is diminished only T ' T . This theoretical re- 

 sult ha* been continued by the admirable experiments 

 of Mr Smeaton, who found, " that the higher the wheel 

 it in proportion to the nhole descent, the greater tciU be 

 Ike effect ;" because, as he remarks, " it depends less 

 upon the impulse of the head, and more upon the gra- 

 vity of the water in the buckets ; and if we consider 

 how obliquely the water issuing from the head must 

 strike the buckets, we shall not be at a loss to account 

 for the little advantage that arises from the impulse 

 thereof, and shall immediately see of how little conse- 

 quence this impulse is to the effect of an overshot 

 wheel." 



If the diameter of the wheel were equal to the whole 

 height of the fall, the water would be laid in the buck- 

 ets without having acquired any velocity ; so that a 

 portion of the power of the wheel would be spent in 

 dragging this inert mass into motion, and also by the 

 impulse of the buckets against the water, which will 

 dash a part of it over Uie w heel I Irnce it is necessary 

 that the difference between the bead of water and the 

 diameter of the wheel should be such, that the water 

 nay acquire in its descent through that space a veloci- 

 ty a little greater than that of the circumference of the 

 wheel. In this view of the subject, the water should 

 fall through a height of { or 5 inches per second, in 

 order to acquire the velocity of the wheel ; and there- 

 fore the diameter of the wheel should be only 3 inches 

 less than the height of the fall. 



The determination of the diameter of an overshot 

 wheel, as given by Borda, Smeaton, Robison, and other 

 authors, is founded upon the assumption, that it never 

 should exceed the height of the fall. Let us suppo.tr 

 that we have a fall of 12 feet, and that the wheel should 

 have a diameter of 1 1 feet according to Borda, then it 

 appear* to us, that a great advantage will be derived 

 from making the wheel 15 fret. Now it is obvious, 

 that the advantage of using the 15 feet wheel is, that 

 we apply Uic water where it will act most perpendicu- 

 larly to the line OD, Fig. 15. or the radius of the 

 wheel, whereas the disadvantage of such a wheel is, 

 that it begins to lose its water much sooner than the 

 small one. We differ in opinion from Itobison when 

 be says, that the loss of power in the latter case exceeds 

 what is gained in the former cue; but we shall admit that 

 it is so, and still maintain the superiority of the 15 feet 

 wheel. When the wheel has a diameter less than the 

 Atnitug* height of the fall, any augmentation of the quantity of 

 water discharged by the mill course is of no use in in- 

 creasing the eiect of the wheel. The issuing water 

 indeed acquires a velocity greater than it usually has, 

 but this additional velocity is injurious to the motion 

 of the wheel instead of being of any advantage. In the 

 case of a 15 feet fall, however, when the water rises I 

 or 2 feet above it* usual level, we have it in our power, 

 by a particular form of the delivering sluice, which 

 will afterward* be described, (seejp. 555,) to introduce 

 this water upon the whet! 1 or a fctt higher up the 



VOL. XI. PART II. 



Ik* liMBM- 



lerofthe 



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Mghtof 



the Uil. 



wheel, so that we are actually enabled to increase the Overshot 

 height of the fall by this quantity. Whel. 



From a series of experiments on overshot wheels, by 

 M. Deparcieux, andpublished in 1754, he has conclud- 

 ed, that most work is performed by an overshot wheel of overshot 

 when it moves slowly, and that the more we retard its wheels, 

 motion by increasing the work to be performed, the 

 greater will be the performance of the wheel. These Depar- 

 experiments were made with a wheel 20 inches in dia- cieux' ex- 

 meter, and having 48 buckets. Cylinders of different I*"" 1 * 1 "*- 

 diameters were placed upon the axle, and the effect of 

 the wheel under different velocities was measured by 

 the height to which it raised a weight suspended to a 

 rope, which was wound round the different cylinders ; 

 and the general result was, that the slower the wheel 

 turns, the greater is the effect, or the height to which 

 the weight is raised. 



In opposition to these results, the Chevalier O*Arcy 

 maintained, that there is a certain velocity when tin- 

 effect is a maximum ; and he has shewn, from a com- 

 parison of Deparcicux's experiments with his own for- 

 mula;, that the wheel never moved with such a small 

 velocity as would have given the maximum effect, and 

 that if he had increased the diameter of his cylinders, 

 he would have found that there was a velocity when 

 the maximum effect began to diminish. 



The experiments of Smeaton afford an excellent con- 

 firmation of tl preceding reasoning. The wheel which 

 he used was 25 inches in diameter. The depth of tin 

 buckets or of the shrouding, was 2 inches, and tlie 

 number of bucket* S&. When it made about -Jo tunu 

 in a minute, the effect was nearly the greatest. Vvlu-n 

 the number of turns was SO, the effect was din.iiii-lu <l 

 ,' 9 th part. \Vht M the number was +0, tli- diinin. 

 was ^th ; when the number wu lets than 18| its mo- 

 tion was irregular ; and when it was loaded so as not 

 to be able to make IS turns, the wheel was overpower- 

 ed by its loaJ. 



" It is an advantage in practice," says Mr Smeaton, Akmrt of 

 " that the velocity of the wheel should not l>e dimmis-h- SOJIOD' 

 ed farther than what will procure some solid advantage **!*" 

 in point of power ; because, cceieru parti>tu, as the mo- m 

 tion is slower the buckets must be made larger ; anil 

 the wheel being more loaded with water, the stress 

 upon every part of the work will be increased in pro- 

 portion. The best velocity for practice, therefore, will 

 be such, as when the wheel here n-ed made about SO 

 tarns in a minute ; that is, when the velocity of the cir- 

 rence is a little more than three feet in a second. 



" Experience confirm*, that thi* velocity of three 

 feet in a second is applicable to the highest overshot 

 wheels a* well a* the lowest ; and all other parts of 

 the work being properly adapted thereto, will pro- 

 duce very nearly the greatest effect possible ; however, 

 this also is certain from experience, that high wheels 

 may deviate farther from this rule before they will lose 

 their power by a given aliquot part of the whole, than 

 low ones can be admitted to do ; for a wheel of 24 feet 

 high may move at the rate of six feet per second, with- 

 out losing any considerable part of its power ; and, on 

 the other hand, I have seen a wheel of 33 feet high, 

 that has moved very steadily and well with a velocity 

 but little exceeding two feet." 



The experiments of the Abbe Bossut afford the Boom'xx. 

 same results. He used a wheel three feet in diameter, perimenu. 

 The height of the buckets was three inches, their width 

 five inches, and their number 48 ; and the canal which 

 conveyed the water furnished uniformly 1194 cubic 

 inches in a minute. When the wheel was unloaded, it 



