The area of the orifice and the velocity of the flow being known, it is easy to calculate the quantity of water discharged in a given time. Thus, suppose the area to be 1 sq.in., and the velocity 20 ft. a second, it is evident that there issues in a second a cylinder or a prism of water 1 sq.in. in section and 20 ft. long, the content of which is cubic inches. In any given time, then, as three minutes (=180 seconds), the discharge is 240X180=43,200 cubic inches.
It has as yet been assumed that the water in the vessel or reservoir is kept constantly at the same height, and that thus the velocity is constant. We have now to consider the case of a vessel allowed to empty itself through an orifice at the bottom. As the surface of the water sinks, the velocity of the discharge diminishes or is retarded; and when the vessel is of the same area from top to bottom, it can be proved that the velocity is uniformly retarded. Its motion follows the same law as that of a body projected vertically upwards. Now, when a motion uniformly retarded comes to an end, the space described is just half what the body would have passed over had it gone on uniformly with the velocity it had at the outset. Therefore, when the vessel has emptied itself in the way supposed, the quantity discharged is hart what would have been discharged bad the velocity been uniform from the beginning.
The "Contraction of the by means of the area of the opening and the velocity thus determined, we calculate the number of cubic feet or of gallons that ought to flow out in a given time, and then measure the quantity that actually does flow, we find that the actual flow falls short of the theoretical by at least a third. In fact, it is only the central part of the jet, which approaches the opening directly, that has the velocity above stated. The outer particles approach from all sides with less velocity; they jostle one another, as it were, and thus the flow is retarded. In conse quence of this want of uniformity in velocity and direction among the component layers of the jet, as they enter the orifice, there takes place what is called a "contraction of the vein" (vena contracta); that is, the jet, after leaving the orifice, tapers, and becomes narrower. The greatest contraction is at a distance from the orifice equal to half its diameter; and there the section of the stream is about two-thirds the area of the opening. It is, in fact, the section of the contracted vein that is to be taken as the real area of the orifice, in calculating by the theory the quantity of water discharged. If the wall of the vessel has considerable thicliness, and the orifice is made to widen gradually inwards, in the proportions of the contracted vein, the stream does not suffer contraction, and the area of the orifice where it is narrowest may be taken as the actual area of discharge.
has as yet been supposed that the issue is by means of a simple opening or hole in the side or bottom of the vessel; but if the flow takes place through a short tube, the rate of discharge is remarkably affected. Through a simple opening, in a thin plate, the actual discharge is only about 64 per cent of the theoretical; through a cylindrical conducting-tube, or adjutage, as it is called, of like diameter, and whose length is four times its diameter, the discharge is 84 per cent. The effect is still greater if the discharge-tube is made conical both ways, first contracting like the con tracted vein, and then widening. The effect of a conducting-tube in increasing the
discharge is accounted for by the adhesion of the water to its sides, which widens out the column to a greater area than it naturally have. It has thus a tendency to form a vacuum iiii.t1to tube, which acts Ow suction on the water in the reservoir, and increases the quantity discharged: The flow is more free if the orifice is in the bottom of the vessel, than in the side on a level with the bottom. If the discharge-tube is made to project inwards beyond the thickness of the walls of the vessel, the velocity is much impeded, owing to the opposing currents produced by the water approaching the opening.
Pipes.—When a conduit pipe is of any considerable length, the water issues from it at a velocity less than that due to the head of water in the reservoir, owing to the resistance of friction. With a pipe, for instance, of 1 in. in diameter, and 30 ft. long, the discharge is only one-half what it would be from is simple orifice of the same diameter. The rate of reduction depends upon the diameter of the tube, its length, the bendings it undergoes, etc. The resistance to the flow of water in pipes does not arise properly from friction, as understood of solids, but from the adhesion of the water to the sides of the pipe, and from the cohesiou of the watery particles among them selves; it makes little difference, therefore, whether an earthenware pipe, for instance, be glazed or not. Large projections form an obstacle; but mere roughness of surface is filled up by an adhering film of water, which is as good as a glaze. The resistance increases greatly with the narrowness of the pipes. Engineers have formulas, deduced in great part from experiment, for the discharge through pipes of given length and diameter. and with a given head; but the subject is too complicated for introduction here. If water flowed in a conduit pipe without friction or other obstruc tion, so that its velocity were always equal to that due to the head of water, there would be no lateral or bursting pressure on the walls of the pipe; and if the pipe were pierced, the water would not squirt out. Accordingly, with a short tube or adjutage, which, instead of obstructing, increases the flow, there is not only no lateral outward pressure on the walls of the tube, but there is actually a pressure inwards. If a hole is made in the wall of a cylindrical adjutage and the one end of a small bent tube is inserted in the hole, while its other end is dipped in a vessel of water, the water will be sucked up the tube, showing the tendency that the adjutage has to form a vacuum. But when the velocity of discharge is diminished by the friction of a long pipe, or by any narrowing, bending, or other obstruction in the pipe, then that portion of the pressure of the head of water that is not carried off in the discharge, becomes a burst ing pressure on the walls of the pipe. This pressure is unequal at different parts of the pipe. At the end where the water issues free and unobstructed, it is next to nothing, and gradually increases towards the reservoir, where it is equal to the difference between the head of water in the cistern, and the head due to the velocity with which the water is actually flowing in the pipe. The principle now explained accounts for the fact, that pipes often burst or begin to leak on the motion of the water in them being checked or stopped.