556 



HYDRODYNAMICS. 



Overshot d'Hydrntlynamique, edit. 1796,- torn. i. chap. xvii. p. 



Wheels. 533 . an j to , n jj cna p xv iii. p. 425. Fcnwick's Four 

 "Y"""*' Essays on Practical Mechanics. Robison, System of' 

 Mechanical Philosophy, vol. ii ; and Ferguson's Lec- 

 tures on Mechanics, &c. vol. ii, Appendix. 



Undershot 

 wheels. 

 PLATE 

 CCCXX. 

 Fig. 7. 



Construc- 

 tion t the 



mill-course, 

 Fig. 8. 



SECT. II. On Undershot Water IVJicch. 



AN undershot water wheel is a wheel with a number 

 of floatboards, or plane surfaces arranged round its cir- 

 cumference for the purpose of receiving the impulse of 

 the water, which is conveyed to the under part of the 

 wheel from an inclined canal. A wheel of this kind 

 of the ordinary construction, is shewn in Plate CCCXX. 

 Fig. 7. where AB is the wheel with 24 floatboards, cd 

 a floatboard receiving the impulse of the water, which 

 moves with great velocity in consequence of having 

 fallen from a considerable height down the inclined 

 mill course MN. The principal points to be attended to 

 in the construction of undershot wheels, are the con- 

 struction of the mill course, the number, form, and po- 

 sition of the floatboards, and the velocity of the wheel 

 in relation to that of the water when the effect is a 

 maximum. The following rules for the construction 

 of mill courses are given in the Appendix to Ferguson's 

 Lectures, vol. ii. 



" As it is of the highest importance to have the 

 height of the fall as great as possible, the bottom of 

 the canal, or dam, which conducts the water from the 

 river, should have a very small declivity ; for the 

 height of the water-fall will diminish in proportion as 

 the declivity of the canal is increased. On this ac- 

 count, it will be sufficient to make AB slope about one 

 incli in 200 yards, taking care to make the declivity 

 about half an inch for the first 48 yards, in order that 

 the water may have a velocity sufficient to prevent 

 it from flowing back into the river. The inclination 

 of the fall, represented by the angle GCR, should be 

 25 50' ; or CR, the radius, should be to GR, the tan- 

 gent of this angle, as 100 to 48, or as 25 to 12; and 

 since the surface of the water S6 is bent from a 6 into 

 ac, before it is precipitated down the fall, it will be 

 necessary to incurvate the upper part BCD of the 

 course into BD, that the water at the bottom may move 

 parallel to the water at the top of the stream. For 

 this purpose, take the points B, D, about 12 inches 

 distant from C, and raise the perpendiculars BE, DE : 

 the point of intersection E will be the centre from which 

 the arch BD is to be described ; the radius being about 

 10j inches. Now, in order that the water may act 

 more advantageously upon the floatboards of the wheel 

 WW, it must assume a horizontal direction HK, with 

 the same velocity which it would have acquired when 

 it came to the point G : But, in falling from C to G, 

 the water will dash upon the horizontal part HG, and 

 thus lose a great part of its velocity ; it will be proper, 

 therefore, to make it move along FH an arch of a circle, 

 to which DF and KH are tangents in the points F and 

 H. For this purpose make GF and GH each equal to 

 three feet, and raise the perpendiculars HI, FI, which 

 will intersect one another in the point I distant about 

 t feet inches and T * ff ths i'roni the points F, and H, 

 and the centre of the arch FH will be determined. 

 The distance HK, through which the water runs be- 

 fore it acts upon the wheel, should not be less than two 

 or three feet, in order that the different portions of the 

 fluid may have obtained a horizontal direction: and if 

 HK be much larger, the velocity of the stream would 

 be diminished by its friction on th bottom of the 



course. That no water may escape between the bot- 

 tom of the course KH and the extremities of the float- 

 boards, KL should be about three inches, and the ex- 

 tremity o of the floatboard n o should be beneath the 

 line HKX, sufficient room being left between o and M 

 for the play of the wheel, or KLM may be formed into 

 the arch of a circle KM concentric with the wheel. 

 The line LM V, called by M. Fabre, the course of im- 

 pulsion (fc coursicr d' impulsion ) should be prolonged, 

 so as to support the water as long as it can act upon 

 the floatboards, and should be about 9 inches distant 

 from OP, a horizontal line passing through O, thf, 

 lowest point of the fall ; for if OL were much less than 

 9 inches, the water having spent the greater part of its 

 force in impelling the floatboards, would accumulate 

 below the wheel and retard its motion. For the sam< 

 reason, another course, which is called by M. Fabre. 

 the course of discharge (le coitrsier de dechargc) should 

 be connected with LMV by the curve VN, to preserve 

 the remaining velocity of the water, which would 

 otherwise be destroyed by falling perpendicularly from 

 V to N. The course of discharge is represented by 

 VZ, sloping from the point O. It should be about Iff 

 yards long, having an inch of declivity in every two 

 yards. The canal which reconducts the water from 

 the course of discharge to the river, should slope about 

 4 inches in the first 200 yards, 3 inches in the second 

 200 yards, decreasing- gradually till it terminates in 

 the river. But if the river to which the water is con- 

 veyed, should, when swollen by the rains, force the 

 water back upon the wheel, the canal must have a 

 greater declivity, in order to prevent this from taking 

 place. Hence it will be evident, that very accurate 

 levelling is necessary for the proper formation of the 

 mill course." 



The general ideas contained in the preceding con- 

 structions appear to have been first suggested by Du 

 Buat, and afterwards fully explained by M. Fabre in 

 his Truitc stir les Machines Hi/drauliques. 



The diameters of undershot wheels must in general 

 be accommodated to the nature of the machinery which 

 they are to put in motion. If a great velocity is necessary, 

 the wheel should for this purpose be made of a less 

 diameter than would otherwise be advisable ; but if a 

 great velocity is not required, the diameter of the wheel 

 ought to be considerable. 



M. Pilot, one of the earliest writers who attended to 

 this subject, recommended that the number of float- 

 boards should be equal to 360 divided by the arch of 

 the circle plunged in the canal, and that their depth 

 should be equal to the versed sine of that arch. The 

 slightest consideration, however, is sufficient to con- 

 vince us that the number of floatboards obtained by this 

 rule is greatly too small. M. Du Petit Vandin, and 

 afterwartls M! Fabre, have, on the other hand, main- 

 tained, that the greatest possible number of floatboards 

 should be used, provided the wheel is not too much 

 loaded by them. 



In Mr Smeaton's model, by which his experiments 

 were performed, the diameter of the wheel was 24. 

 inches, and the number of floatboards 24. When the 

 number was reduced to 1-2, a diminution in the effect 

 was produced on account of a greater quantity of wa- 

 ter escaping between the floats and the floor; " but a 

 circular sweep being adapted thereto of such a length 

 that one float entered the curve before the preceding- 

 one quitted it, the effect came so near to the former, as 

 not to give hopes of advancing it, by increasing the 

 number of floats beyond 24 in this particular wheel/' 



('onstror- 

 tion of the 

 mill-course, 

 PLATE 

 CCCXX. 

 Fig. 8. 



Number of 

 floatboards. 



Smeaton's 

 experiment 

 on the nuni 

 her of float 

 boards. 



