558 



HYDRODYNAMICS. 



X X V t,= xV 



Smcaton's 

 experi- 



Undershot 



Now this effect is to be a maximum, and therefore its 

 differential must be equal to 0, that is, v being the va- 

 riable quantity, V dv 2v dv = 0, or 2vdv =Vdv. 



y 

 Dividing by dv, we have 2u = V, andw = , that 



is, the velocity of the wheel will be one-half the velo- 

 city of the fluid when the effect is a maximum. 



This has been amply confirmed by the experiments 

 of Mr Smeaton. " The velocity of the stream (says 

 nents. nej p_ 77^ varies at the maximum between one-third 

 and one-half that of the water; but in all the cases in 

 which most work is performed in proportion to the wa- 

 ter expended, and wliich approach the nearest to the 

 circumstances of great works, when properly execut- 

 ed, the maximum lies much nearer to one-half than 

 one-third, one half seeming to be the true maximum, if 

 nothing were lost by the resistance of the air, the scatter- 

 ing of the water carried up by the wheel, &c. all 

 which tend to diminish the effect more at what would 

 be the maximum if these did not take place than they 

 do when the motion is a little slower." Smeaton con- 

 siders 5 to 2 as the best general proportion. 

 5 A result, nearly similar to this, was deduced from 

 the experiments of Bossut. He employed a wheel 

 whose diameter was three feet. The number of float- 

 boards was at one time 48, and at another 24, their width 

 being five inches, and their depth six. The experi- 

 ments with the wheel, when it had 48 floatboards, 

 were made in the inclined canal, in which the velocity 

 was 300 feet in 27 seconds. The experiments with 

 the wheel, when it had 24 floatboards, were made in a 

 canal, contained between two vertical walls, 12 or 13 

 feet distant. The depth of the water was about seven or 

 eight inches, and its mean velocity about 2740 inches 

 in 40 seconds. The floatboards of the wheel were im- 

 mersed about four inches in the stream. 



As the effect of the machine is measured by the pro- 

 duct of the load raised, and the time employed, it will 



appear, by multiplying the second and third columns, Undershot 

 that the effect was a maximum when the load was 34^ Wheels. 

 pounds, the wheel performing 20|-J revolutions in 40 ^""Y"^ 

 seconds. By comparing the velocity of the centre of 

 impression computed from the diameter of the wheel, 

 and the number of turns which it makes in 40 seconds, 

 with the velocity of the current, it will be found that 

 the velocity of the wheel, when its effect is the great- 

 est possible, is nearly two-fifths .that of the stream; 

 the very same ratio which Smeaton has given. Fronj 

 the two last columns of the Table, where the effect is a 

 maximum when the load is 60 pounds, the same con- 

 clusion may be deduced. 



The following are the other results which Mr Smea- gineatoa'8 

 ton deduced from his experiments. He found, that in results. 

 undershot wheels, the power employed to turn the 

 wheel is to the effect produced as 3 to 1 ; and that the 

 load which the wheel will carry at its maximum, is to 

 the load which will totally stop it as 3 to 4. The same 

 experiments inform us, that the impulse of the water 

 on the wheel, in the case of a maximum, is more than 

 double of what is assigned by theory, that is, instead of 

 four-sevenths of the column, it is nearly equal to the 

 whole column. In order to account for this, Mr Smea- 

 ton observes, that the wheel was not, in this case, pla- 

 ced in an open river, where the natural current, after 

 it had communicated its impulse to the float, has room 

 on all sides to escape, as the theory supposes ; but in a 

 conduit or race, to which the float being adapted, the 

 water could not otherwise escape than by moving along 

 with the wheel. He likewise remarks, that when a 

 wheel works in this manner, the water, as soon as it 

 meets the float, receives a sudden check, and rises up 

 against it like a wave against a fixed object ; insomuch, 

 that when the sheet of water is not a quarter of an inch 

 thick before it meets the float, yet this sheet will act 

 upon the whole surface of a float, whose height is three 

 inches. Were the float, therefore, no higher than the 

 thickness of the sheet of water, as the theory supposes, 

 a great part of the force would be lost by the water 

 dashing over it. Mr Smeaton likewise deduced, from 

 his experiments, the following maxims. 



1. That the virtual or effective head being the same, Smeaton'e 

 the effect will be nearly as the quantity of water ex- maxims. 

 pended. 



2. That the expense of water being the same, the 

 effect will be nearly as the height of the virtual or ef- 

 fective head. 



3. That the quantity of water expended being the 

 same, the effect is nearly as the square of the velocity. 



4. That the aperture being the same, the effect will 

 be nearly as the cube of the velocity of the water. 



Undershot Wheel moving at Right Angles to the Stream. 



Undershot wheels have sometimes been constructed undershot 

 like windmills, having large inclined floatboards, and wheel at 

 being driven in a plane perpendicular to the direction right anglw 

 of the current. Albert Euler, who has examined theo. to the 

 retically this species of water wheel, concludes that stream * 

 the effect will be twice as great as in common under- 

 shot wheels, and that in order to produce this effect, the 

 velocity of the wheel, computed from the centre of im- 

 pression, should be to the velocity of the water as ra- 

 dius is to thrice the sine of the inclination of the float- 

 boards to the plane of the wheel. When the inclina- 

 tion is 60, the ratio will be as 5 to 13 nearly, and 

 when it is 30, it will be nearly as 2 to 3. In this kind 

 of wheel, a considerable advantage may also be gained 

 by inclining the floatboards to the radius. In this case, 



