Chap. XVII.

THEORY OF THE MOTION OF FLUIDS.

1145

Counterweight for 30 turns, without water, 2 oz. in the scale ; the area of the head was105-8 square inches ; weight of the empty scale and pulley 10 oz.; circumference of thecylinder 9 inches; circumference of the water-wheel 75 inches.

The circumference of the wheel 75 inches, multiplied by 86 turns, gives 6450 inches forthe velocity of the water in a minute, of which will be the velocity per second, equal to107*5 inches, or 8*96 feet, the head being 15 inches, and the vertical or effective head,which differs from the real head by the velocity lost by the fluid in issuing through theaperture of the sluice, and the friction of the channel. The area of the head being 105 8inches, this multiplied by *579, the weight in avoirdupois ounces of a cubic inch of water,gives 61 -26 ounces for the weight of as much water as is contained in the head upon 1 inchin depth, ^ of which is 3'83 lbs. ; this multiplied by the depth, 21 inches, gives 80*43 lbs.or the value of twelve strokes; and by proportion, 39£, the number made in a minute willgive 264*7 lbs. as the weight of water expended in a minute: 264*7 lbs. of water, havingthen ascended through a space of 15 Inches in a minute, the product of these two numbers,3970, will express the power of the water to produce mechanical effects, which upon thiswheel were as follows: —

The velocity of the wheel at the maximum was thirty turns in a minute ; this multipliedby 9 inches, the circumference of the cylinder, makes 270 inches; but as the scale washung by pulley and double line, the weight was only raised 135 inches.

lbs. oz.

The weight in the scale at the maximumWeight of the scale and puLley -Counterweight of the scale and pulley

8 00 100 12

Sum of the resistances

9 6

or 9*375 lbs.

Now, as 9*375 lbs. are raised 135 inches, these two numbers multiplied together produce1266, the effect produced as a maximum ; so that the proportion of the power to the effectis as 3970 ; 1266, or as 10 : 3*18. Though this is the greatest single effect by the impulse ofthe water upon an undershot wheel, yet, as the whole power of the water is not exhaustedthereby, this will not be the true ratio between the power of the water and the sum of allthe effects producible therefrom ; for as the water must necessarily leave the wheel witha velocity equal to its circumference, it is clear that some part of the power of the w*atermust remain after quitting the wheel. The velocity of the wheel at the maximum is 30turns in a minute, so that its circumference moves at the rate of 3*123 feet per second,which answers to a head of 1 *82 inches; this being multiplied by the expenditure of waterin a minute, viz. 264*7 lbs., produces 481 for the power remaining in the water after ithas passed the wheel; this being, therefore, deducted from the original power, 3970, leaves3489, which is that portion of the power spent in producing the effect 1266, which is tothe greatest effect producible thereby, as

3489 : 1266 10 : 3*62, or as 11 to 4.

The velocity of the water striking the wheel has been determined as equal to 86 circum-ferences of the wheel per minute, and the velocity of the wheel at the maximum to be 30;the velocity of the water will, therefore, be to that of the wheel, as 86 to 30, or as 10 to 3*5,or 20 to 7 : the load of the maximum was equal to 9 lbs. 6 oz.; and the wheel ceasedmoving with 12 lbs. in the scale, to which the weight of the scale was added, viz. 10 ounces;the proportion, therefore, will be nearly as 3 to 4 between the load of the maximum, andthat by which the wheel is stopped. The velocity of the wheel in relation to the water isproved to be greater than J the velocity of the water ; yet the impulse of the water, in thecase of a maximum, is more than double that assigned by theory, viz. instead of £ of thecolumn, it is nearly equal to the whole column: the power of the water is equal to thevelocity and pressure due to the effective head. The resistance reduced to the circum-ference of the wheel is equal to the weight raised, added to the friction of the machinewhen raising that weight, multiplied by the velocity of the wheel. The maximum ofuseful effect takes place when the greatest weight is raised with the greatest velocity : inthe case experimented upon, to prove this, the wheel was not placed in an open river, wherethe natural current, after it has communicated its impulse to the float, has room on allsides to escape, as the theory supposes.

After numerous experiments, Smeaton ascertained that the vertical or effective headbeing the same, the effect will be nearly as the quantity of water expended; and that theexpense of water being the same, the effect will be nearly as the height of the vertical oreffective head, and the effect nearly as the square of its velocity : the aperture being thesame, the effect will be nearly as the cube of the velocity of the water.

Bossut’s experiments on undershot water-wheels confirm many of Smeaton ’s results: hetried their effect in a close canal, as well as in an open stream, varying in both cases the