ing a pump, 4. Id. working 12 hours, and raising plaster, 1684 842 1185 595 } 2948 675 1560 Doubtful. 800 5974 dynamical units. 9 SECT. II. On the Force of Water, and the mode of applying it to drive Machinery. In employing water as the first mover of machinery, it is applied to the circumference of wheels, from the axis of which the power is immediately conveyed to the other parts of the machine. Water-wheels derive their names from the different ways in which the power of the water is applied to them, and are divided into, 1. Undershot wheels. 2. Overshot wheels. 3. Breast wheels; and 4. Wheels driven by the re-action of water; which we shall describe in their order. 1. On Undershot wheels. An undershot water-wheel is a wheel having a number of plane surfaces called float-boards projecting from its circumference, for the purpose of receiving the impulse of the water, which is delivered by a sloping canal, and with great velocity, upon the under part of the wheel. 9 This result is deduced from the fact, that a horse can draw 140 pounds with the velocity of 200 feet in a minute. For farther information on this subject, see the Edinburgh Encyclopædia, Art. Mechanics, vol. xiii, p. 560. This kind of wheel is represented in Plate I, Fig. 1, where W w is the wheel with 24 float-boards, and no one of the floatboards receiving the impulse of water which has acquired great velocity by descending the inclined mill-course A B D F K M. In the erection of undershot wheels, the principal points to be attended to relate to the construction of the mill-course, the size of the wheel, the number, form, and position of the float-boards, and the relation between the velocity of the water and that of the wheel, in order to produce a maximum effect; but as these wheels are commonly applied to mills for grinding corn, we shall introduce under this section a description of corn-mills. On the millcourse. 1 On the Construction of the Mill-Course. As it is of 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. Plate I. On this account, it will be sufficient to make A B Fig. 1. slope about one inch 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 pre vent it from flowing back into the river. The inclination of the fall, represented by the angle G C R, should be 25° 50′; or CR, the radius, should be to G R the tangent of this angle, as 100 to 48, or as 25 to 12; and since the surface of the water Sb is bent from ab into a c, 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 B D is to be described; the radius being about 10 inches. Now, in order that the water may act more advantageously upon the floatboards of the wheel W W, 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 H G, and thus lose a a great part of its velocity; it will be proper, therefore, to make it move along F H, an arch of a circle to which D F and KH are tangents, in the points F and H. For this purpose, make G F and GH each equal to three feet, and raise the perpendiculars HI, FI, which will intersect one another in the point I, distant about 4 feet 9 inches and 4-10ths from the points F and H, and the centre of the arch F H will be determined. The distance H K, through which the water runs before 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 the bottom of the course. That no water may escape between the bottom of the course KH and the extremities of the float-boards, K L should be about 3 inches, and the extremity o of the float-board no should be beneath the line HK X, sufficient room being left between o and M for the play of the wheel, or K L M may be formed into the arch of a circle K M concentric with the wheel. The line LMV, called by M. Fabre the course of impulsion (le coursier d'impulsion), should be prolonged, so as to support the water as long as it can act upon the float-boards, and should be about 9 inches distant from O P, a horizontal line passing through O the lowest point of the fall; for if O L 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 same reason, another course, which is called by M Fabre, the course of discharge (le coursier de decharge), should be connected with L M V, 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 V Z, sloping from the point 0. It should be about 16 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 conveyed should, when swoln 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. In order to find the breadth of the course of discharge, multiply the quantity of water expended in a second,' measured in cubic feet, by 756, for a first number. Multiply the square root of d K (d K being found by subtracting OK, or PR, each equal to a foot, from dO or b R, the height of the fall) by OL, or of a foot, and also by 1000, and the product will be a second number. Divide the first number by the second, and the quotient will be nearly the least breadth of the course of discharge. If the breadth of the course, thus found, should be too great or too small, the point L has been placed too far from O, or too near it. Increase, therefore, or diminish OL; and having subtracted from d O or b P, the quantity by which O K is greater or less than a foot, repeat the operation with this new value of d K, and a more convenient answer will be found. The preceding rule will give too large a breadth to the course, when the expense of water is great, and the height of the fall inconsiderable. But the course of discharge ought always to have a very considerable breadth, which should be greater than that of the course of impulsion, that the water having room to spread, may have less depth; and that a greater height may be procured to the fall, by making O L, and consequently OK, as small as possible; for the breadth of the course is inversely as OL, that is, it increases as O L diminishes, and diminishes as it increases. The reader may suppose that this rule still leaves us to guess at the breadth of the course of discharge; but, from the purposes for which it is used, it is easy to know when it is excessively large or small; and it is only when this is the case, that we have any occasion to seek for another breadth, by taking a new value of O L. The section of the fluid at K should be rectangular, the breadth of the stream having a determinate relation to its depth. If there is very much water, the breadth should be triple the 1 The quantity of water expended in a second may be found pretty accurately by measuring the depth of the water at a (A B, the bottom of the canal, being nearly horizontal, and its sides perpendicular), and the breadth of the canal at the same place. Take the cube of the depth of the water in feet, and extract the square root of it. Multiply this root by the breadth of the canal, and also by 507. Divide the product by 100, and the quotient will be the expense of water in a second, measured in cubic feet. This rule is founded on the formula, x = = 5.07 bXd; where x is the quantity of water expended in a second, b the breadth of the canal, and d its depth. depth; if there is a moderate quantity, the breadth should be double the depth; and, if there is very little water, the breadth and depth should be equal. That this relation may be preserved, the course at the point K must have a certain breadth, which may be thus found:-Divide the square root of d K (found as before) by the quantity of water expended in a second, and extract the square root of the quotient. Multiply this root by .623, if the breadth is to be triple the depth; by .515, if it is to be double; and by .364, if they are to be equal, and the product will be the breadth of the course at K. The depth of the water at K is therefore known, being either one third, or one half of the breadth of the course, or equal to it, according to the quantity of water furnished by the stream. Plate I. Fig. 1. In Fig. 1, b P is called the absolute fall, which is found by levelling. Draw the horizontal lines b d, PO; dO will thus be equal to b P, and will likewise be the absolute fall. The relative fall is the distance of the point d from the surface of the water at K, when the depth of the water is considerably less than d K, but is reckoned from the middle of the water at K, when dK is very small. The relative fall, therefore, may be determined by subtracting O K, which is generally a foot, from the absolute fall d O, and by subtracting also either the whole or one half of the natural depth of the water at K, according as d Kis great or small in proportion to this depth. Breadth of fall. The next thing to be determined is the breadth of the course at the top of the fall B, and the breadth the course at of the canal at the same place. To find this, multi- the top of the ply the quantity of water expended in a second by 100, for a first number; take such a quantity as you would wish, for the depth of the water, and, having cubed it, extract its square root, and multiply this root by 507, for a second number; divide the first number by the second, and the quotient will be the breadth required. The breadth, thus found, may be too great or too small in relation to the depth. If this be the case, take one half of the breadth, thus found, and add to it the number taken for the depth of the water; the sum will 2 The depth of the water, here alluded to, is its natural depth, or that which it would have if it did not meet the float-boards. The effective depth is generally two and a half times the natural depth, and is occasioned by the impulse of the water on the float-boards, which forces it to swell, and increases its action upon the wheel. |