94 CHAPTER II. ON THE FLOW AND DISCHARGE OF WATER, AND THE ESTIMATION OF WATER POWER. In the present chapter it is proposed to enter only so far into those questions of Hydrodynamics which relate to the measurement of the discharge of water, and the estimation of water power, as it is necessary they should be understood by the practical millwright, in order that he may be at no loss in comparing the efficiency of various forms of water machinery, calculating their power, and proportioning them to their position and their work. For minute and accurate mathematical investigations, the reader is referred to special treatises on hydrodynamics, in which the subject is treated from another point of view. From the nature of a non-elastic fluid such as water, in which the particles are free to move over one another without friction, the following relations hold between pressure, velocity and discharge. 1st. The pressure P upon a unit of area at the depth h beneath a fluid surface is equal to the weight of a column of the liquid h units high; that is, if w be the weight of a unit of volume, P = wh...(1). And therefore the pressure on a horizontal surface of a units area wha. 2nd. The velocity with which a fluid flows from a small orifice at the depth h beneath the surface, is the same as the velocity it would have acquired in falling freely the same distance under the action of gravity, if we neglect those causes of retardation to be considered presently. If we take v = mean velocity of the effluent water; h = mean depth of orifice beneath the surface, or, in other words, the head of fluid; g = 32.1908 = the velocity generated in a falling body in one second; then by the laws of accelerating motion, v = √ 2 gh*,... (2) that is, the theoretical velocity of effluent water is equal to the square root of 64-38 times the mean head; understanding by mean head the head measured from the centre of the orifice. Thus we have in the following table the theoretical velocity at various heads. 3rd. The quantity of water which issues from an orifice at a depth h beneath the surface of a fluid is equal to the area of the orifice multiplied by the velocity of the effluent water, that is, neglecting the diminution from the venâ contractâ to be mentioned shortly. Let Q units of volume discharged per second a = area of orifice v = velocity of effluent water Q = a v = a √ 2 g h ... (3). And in t seconds q ta vt will be discharged. Where q is called the theoretical discharge, and is found by multiplying the area of the orifice in feet by the velocity of the effluent water in feet per second, found as above. 4th. If the orifice instead of opening freely into the air, as supposed above, opens into another reservoir of fluid, we must substitute in the above equations the difference of level of fluid *Or in feet v= = 8·03/h. in the two reservoirs for the head above the centre of the orifice. Let h' be the head above the centre of the orifice in the higher reservoir, and h" in the lower; then the effective head h=h' — h' h"... (4). 5th. If the water escape by a rectangular notch instead of an orifice, that is, an aperture such that the upper level surface of the water does not come in contact with the sides of the vessel, falling freely in the air, the theoretical discharge is two-thirds of the area of the effluent vein multiplied by the velocity of efflux; or, more accurately, if h = head of water, b = breadth of notch. q=fb.h. √2 g h ... (5). We must next examine certain properties of fluid motion which cause the actual or effective discharge to differ materially from the theoretical discharge given in the above equations, although in a constant ratio, so that the one may always be calculated from the other. 1st. Thick-lipped orifices or mouth pieces. For smooth orifices, the length of which is about twice or three times the smallest diameter, the actual does not widely differ from the theoretical discharge. The velocity of the effluent current is, however, never so great as that in equation (2), but is diminished for a constant ratio for each kind of orifice, and the discharge is less in the same proportion. For a simple cylinFig. 96. Fig. 97. Fig. 98. Fig. 98. drical tube, fig. 96, of about 14 diameters in length, the velocity of the effluent water is equal to 0.8 v = 0·8 2 g h and the actual discharge a x 0.8 x v = 0·8 Q. = Where the interior angle of the tube is rounded, as in fig. 98, the velocity amounts to as much as 0.96 v to 0.98 v, and hence the discharge to 0.96 Q to 0.98 Q, where Q as before is the theoretical discharge given by the above formulæ. This constant ratio is called the coefficient of velocity. For Hence we have this rule for determining the quantity of water discharged by a thick-lipped orifice; seek first in Table I. the velocity corresponding to the given head of water, measured from the centre of the orifice; multiply this velocity by the area of the orifice in square feet, and the product will be the theoretical discharge in cubic feet per second. the actual discharge, this must be multiplied by 0.7, if the orifice be of the form of fig. 97; by 0.8, if the orifice be of the form of fig. 96; and by 0.97 if it be of the form of fig. 98. The importance of the form of orifice is manifest, and hence a trumpet mouth* should be employed in all water pipes, wherein a maximum discharge is desirable, the quantity being increased whichever way the trumpet mouth is turned, whether as in fig. 98 or 99, but most in the former case. For conical converging tubes, d'Aubuisson found the coefficient of efflux to vary from 0.829 to 0.946 as the lateral convergence increased from 0° 0' to 13° 24', and from 0.946 to 0.847 as the convergence increased from 13° 24′ to 48° 50'; the area of the orifice being measured at the small extremity. For tubes which at first converge and then diverge, so as to take the form of the fluid vein, the coefficient of discharge is 1.55, that is, of course, taking the minimum area of the tube. 2nd. Thin-lipped orifices, the fluid escaping freely into the air. With orifices of this nature, the fluid vein contracts very remarkably at a short distance beyond the orifice, and the discharge is diminished in the ratio of the least area of the vein to that of the orifice. This contraction amounts to five-eighths of the area of the orifice in most cases, and hence the actual discharge is scarcely more than five-eighths that estimated by equation (3). Putting m = the coefficient of contraction, we have the actual discharge from an orifice = m q=ma √2gh. The velocity of the effluent vein is also diminished in a slight degree, perhaps by three or five per cent., but it will be most convenient to combine the coefficients of contraction and velocity together, and to call m the coefficient of discharge or ratio of actual to theoretical discharge, PART I. * As for instance in the reservoir drawing fig. 87. H A very large number of experiments have been made upon the values of the coefficient m for various forms of orifices, the most important of which we owe to Michelotti, Castel, Bidone, Bossut, Rennie, and others. But by far the most important and complete are those conducted by MM. Poncelet and Lesbros under the auspices of the French government, and all interested in hydraulic investigations must feel indebted to them for the skill, perseverance, and accuracy with which they have registered so large a body of results. These determinations go to show that the value of the coefficient of discharge ranges between 0.58 and 0.7,* being greater for small orifices and small velocities and less for large orifices and high velocities.† For heads of three and four feet and upwards, the coefficient of discharge may be taken at 0.6, Mr. Rennie's results give the following values of m. For more accurate calculations I have abridged the following tables of M. Poncelet's results from the Aide-Mémoire' of M. Morin, reducing the measures to the English standard. TABLE II.-COEFFICIENTS OF DISCHARGE OF VERTICAL RECTANGULAR ORIFICES, THIN-LIPPED, WITH COMPLETE CONTRACTION. THE HEADS OF WATER MEASURED AT A POINT OF THE RESERVOIR WHERE THE LIQUID WAS PERFECTLY STAGNANT. |