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THEORY AND PRACTICE OF ENGINEERING.

Book II.

To facilitate the calculations upon the ex-penditure of water, Belidor has furnished someexcellent tables relative to the uniform velocity ofthe falls during a second of time, constructed inarithmetical progression. The parabola A D C, theaxis of which, A B, is supposed to be 15 feet, andits greatest ordinate, B C, 30 feet, was the figureadopted: the ordinates E F, G H, &c., express theexpenditure at the several heights.

For the quantity of water discharged over a weir3\Ir. Eytelwein represented the velocity, whichvaries as the square root of the height, by the or-dinates of a parabola, and the quantity of waterdischarged by the area of a parabola two* thirds ofthat of the circumscribing rectangle: hence thequantity of water discharged may be found bytaking two-thirds of the velocity due to the mean Fig. 1711.

height, and allowing for the contraction of the vein. *

On the Manner of estimating the ff'aste caused hy the Edges of Orifices. — As water flowsmore swiftly towards the centre than at the edges of orifices, being retarded by the fric-tion they occasion, less will issue from an orifice during a specified time, than would hethe case if all the threads had uniform velocity ; consequently the expenditure statedin the preceding calculations is greater than in actual experience, and this difference in-creases as the orifices become less, because the circumferences ofeircles being to each otheras the diameters, while their superficies are as the square of these diameters, the smallorifices, having more circumference in proportion than large ones, the velocity of water withrelation to its quantity will be proportionally retarded.

To find this ratio, we must take the squares of the diameters for the superficies of theorifices, and the sides of these squares as their circumferences: thus railing a the diameter

of the lesser, and b that of the greater, we shall have — for the ratio of the circuit of the

aa

first to its superficies, and the ratio of the circuit of the second to its superficies, whichbb

reduces it to - and ^; whence we have -,a b a

since - = or ab = ab , which shows thata b

the ratio of the circuit of the first to its superficies, is to the ratio of the circuit of the secondto its superficies, as the diameter of the second, to the diameter of the first.

It follows that the ratio of the decrease of the first orifice to its natural expenditure willbe to the ratio of the decrease of the second to its natural expenditure, as the diameter ofthe second is to the diameter of the first. By natural expenditure it must be understoodthat which is found by rule, without considering accidental circumstances, and by waste,the excess of the natural expenditure above the effective, found by experiment.

M hen the ratio of the waste to the natural expenditure of any orifice is determined byexperiment, we shall have the ratio for any other orifice thus : as the diameter of the givenorifice, is to the diameter of that of the experiment, so the ratio of the waste to the naturalexpenditure found in the experiment, is to a fourth term, which will give that required.

M. Mariotte states in his “ Traite du Mouvement des Eaux” that he lias proved by anumber of experiments that from a horizontal orifice, three lines in diameter, 13 feet aboNethe surface of the water, 14 pints or 28 lbs. issued in one minute.

Let the circle AB represent an orifice, divide its diameterat the point C in the ratio of the effective expenditure to thewaste : calling A B, a, and C B, b , the square D B (a a) willexpress the natural expenditure, the rectangle F B ( ah ) thedecrease, and the rectangle DC ( aa — bb) the effective ex-penditure. In like manner, calling d the diameter of anotherorifice, dd will express its natural expenditure, and as thedecreases are to each other in the ratio of the diameter, weshall have a, d .* a 6, bd; whence we have dd—bd for the ex-pression of the effective expense of the second orifice.

If we take aa — ab and dd — bd to express the ratio of thetwo orifices, and Mm to express that of their expenditure, weshall have M, m * aa —at, dd — bd, when the times areequal, and the reservoirs the same height. Calling V thevelocity of the water at the orifice, whose expenditure is ex-pressed by M and T, the time of flowing, u the velocity of the

water of the second reservoir, whose expenditure is expressed

Fig. 1712,