The three varieties of water-wheels already noticed, are the only ones gene-ially admitted into practice, and they do not admit of much improvement, since their principles must always remain the same. The over-shot wheel has, perhaps, been brought nearer to perfection than any of the others, by the contrivance of Peter Nouaille, Esq. who, in a mill that he has near Seven Oaks, in Kent, has caused the water to revert back again from the top of the wheel, instead of passing over it; and in this way a much greater portion of the circumference of the wheel is brought into action than is generally the case. Other improvements or variations in the form and construction of water-wheels, have been contrived by Mr. Besant, Mr. Smart, Mr. Perkins, and others, which will be found described in the Transactions of the Society for the Encouragement of Arts, Manufactures, and Commerce; the object of them principally being to obtain as much force as possible from the water, by arranging the forms of the buckets or float-boards, in such manner that they may receive the greatest impulse or retain the greatest quantity of water, which is of great importance, particularly in the construction of under-shot wheels, which act by the impulse of the water alone.
The over-shot wheel depends entirely on the weight of the water delivered into its buckets, which ought, therefore, to be as capacious as they can conveniently be made, - not only that they may contain as much water as possible, but allow ample room for the discharge of the air that will be thrown into them with the water, as well as for the delivery of that water, when done with. From the nature of a water-wheel, it will be evident, that if it had no work to perform, or resistance to overcome, it would move with the same velocity as the stream that drives it; while, on the contrary, if it was loaded with a quantity of resistance, equal to the power of the stream, it could not move at all: hence, every degree of resistance between these extremes, will produce its proportionate retardation of the wheel; and from accurate experiments which have been tried, it has been determined, that an under-shot wheel does its maximum quantity of work, when its circumference moves with between one-half and one-third of the velocity of the stream that drives it.
The over-shot wheel cannot be so influenced by the velocity of the water, because it requires all its buckets or cells to be filled in succession; and Mr. Smeaton has determined, that the best velocity to effect the above purpose, is three feet in a second. Having, therefore, previously determined the quantity of water which the stream will deliver in a given time, it becomes a matter of easy calculation to determine the length and capacity of the buckets which shall be capable of carrying off the water at that velocity. Thus, for example, if the stream is found to deliver ninety-six gallons per second, and it is determined to make the buckets on the wheel six inches apart from one partition to another, and fifteen inches deep, then six such buckets will be contained in every three feet of the wheel; therefore, ninety-six gallons must be divided by six buckets, which gives sixteen gallons for the contents of each. It will, therefore, only remain to be determined, how long a vessel of six inches wide, and fifteen inches deep, must be, to contain sixteen gallons, and this will, of course, give the necessary width of the wheel, while the number of buckets must depend upon the circumference, which is always limited by the diameter, beingthe extreme height, (if necessary,) that can be obtained in the fall of water; for the larger the wheel, the greater will be the power derived from it, provided a due velocity can be maintained at the same time; because the power of water on wheels, is as the square root of the height it falls through, it being regulated by the same laws as apply to solid bodies in falling.
The power of every wheel, of course, depends upon the quantity of water thrown upon it, and the height from which it has to fall; but as every bucket must be filled, or every float-board struck by the water in succession, so, of course, if the wheel is too large, it will move too slowly for the purpose for which it is intended; and, in this case, the speed must be raised by cog-wheels within the mill, which, on the common principle of mechanics, must dissipate the power intended to be gained by the magnitude of the water-wheel. Hence, great attention should be paid in the construction of mills, to let the size of the water-wheel be well-proportioned, not only to the velocity of the stream, but to the speed of the work it is required to perform; and this may always be accomplished without waste or difference of power, by using a wider wheel of small diameter, -where great speed is necessary, or a narrow wheel of great diameter, when this is not essential. In every case, the full power of a stream should be taken advantage of, in the first erection of a mill, because it is a troublesome and expensive operation to increase the power of a mill, when once built; and power is always valuable.
Mr. Banks, in his excellent Treatise upon Mills, gives many useful practical rules; from amongst which the following is selected. Being simple, it may prove useful for determining the quantity of water that will flow through a sluice or pen-stock upon a wheel, with sufficient accuracy for most purposes, because the whole motion of a stream must not be taken when it is principally dammed or stopped, and only permitted to flow through a small orifice, to produce mechanical effect.
Measure the depth from the surface of the water to the centre of the orifice of discharge, in feet, and extract the square root of that depth; multiply it by 5.4, which will give the velocity in feet per second, and this, multiplied by the area of the orifice (also in feet,) will give the number of cubic feet which will flow through in a second. From knowing the quantity of water discharged, and the height of fall, not only the size of the wheel, but its extent of power may be calculated; for, in the undershot wheel, the power is to the effect nearly as 3 to 1; while in the over-shot wheel it is double, or as 3 to 2.