mercury: the particles situated immediately in front of the orifice, and in which it is necessary to create a certain velocity, are, it is true, fourteen times more dense than those of water, and they, consequently, oppose to motion a resistance fourteen times greater than it would do; but the mass also which presses upon these particles, and produces the velocity of exit, being greater in the same proportion, gives a motive force fourteen times greater. Thus a compensation exists, and the velocity impressed remains the same; and in like manner it may be proved for a fluid lighter than water.

11. The proposition that has now been laid down with respect to the velocity of water issuing through an orifice is equally true in cases when the discharge takes place in vacuo, as when in the atmosphere, the velocity is always the same, with the same head, whatever be the pressure upon the free surface of the water in the vessel, provided the jet of water at its exit from the orifice be subject to an equal exterior pressure. But the velocity will be very different from that due to H if the pressures be not equal upon these two surfaces.

If the pressure be greater against the orifice at A than upon the free surface of the water BC (Fig. 2), then the excess of the former above that on the free surface must be less than that of a column of the fluid whose height is the vertical distance of the orifice A below the surface BC, otherwise there could be no discharge. Let us, then, take an horizontal plane DE below the plane BC, and at such distance from it that the weight of the column of the water contained between the two planes, and whose base is the unit of surface, may be equal to the excess of pressure at A, of which we are speaking,—the water in the vessel having but a very slight degree of motion on account of the relatively small area of the orifice, which is always understood to subsist; and therefore we may assume the pressures to be transmitted as if the contained liquid was in equilibrium. The pressure, then, which exists upon any point in the plane DE will be equal to that upon any point in BC, plus the supposed excess of pressure against the exterior of the orifice; and, therefore, the pressure will be the same upon the plane DE and that against the orifice at A. The liquid below the plane DE is then in the same condition as if that contained between BC and DE were removed, and the free surface and exterior of the orifice were under equal pressures; and thus the same formula will represent the velocity: V = √ 2g (H − h1),

h, denoting the depth of the plane DE below BC.

12. If the exterior pressure on A were less than that upon the surface BC, we may conceive the excess of pressure upon it to be produced by a liquid of the same specific gravity as that in the vessel, applied above BC and terminating in a free surface D'E', situated at such height that the vertical distance represents as before the column of the liquid whose pressure equal to the excess of the pressure on BC above that against the orifice at A. The flow, then, will take place with the same velocity as if the free surface of the liquid, instead of being in the plane BC, and supporting this excess of pressure, were at D'E' and supported the same pressure as the orifice at A; the formula will then be—

V = √2g (H+h2)

if we take h2 equal to the vertical distance of D'E' above BC. We see thus that a diminution or augmentation of the pressure upon the free surface of the liquid in the vessel, without any change in that against the orifice at A, causes a corresponding diminution or augmentation in the velocity of the issuing fluid, and, on the contrary, that a diminution or augmentation of the pressure against the orifice, without any change in that upon the free surface, causes a corresponding augmentation or diminution in this velocity.

EXAMPLE. The condenser of a low-pressure steam engine offers an example of the second case; for, let us suppose a vacuum of 25 inches of mercury to be maintained, and that the head of water in the cistern (Fig. 3) supplying the jet of cold water which effects the condensation, were 2 feet above the point at which it enters this partial vacuum, then the actual head producing the flow is 2+28.25 30.25 feet, for pure mercury being 13.56 times heavier than water, we have the height of a column of water which would balance that of 25 inches of mercury, equal to 25 x 13.56339 inches, or 28.25 feet.

=

The self-acting contrivance for supplying the feed-water to low-pressure boilers (Fig. 4) comes under the first case: the pressure being supposed 5 lbs. per inch above the atmosphere, it is required to place the cistern of the supply so high, that on the opening of the valve a, by the float b descending below the proper level, the water may enter against the pressure of the steam. Now, as the cubic foot of water weighs 62.5 lbs., a column

6.25

144

I foot high and I square inch base weigh = 0.434 lbs., and, therefore, the height of the column of water to balance any given pressure expressed in pounds per square inch is found

by dividing that number by 0.434, in this case, 50.434=11.52 feet: this gives exact equilibrium; the additional head in order that it may enter with due rapidity (from 2 to 4 feet generally) will depend upon the consumption of the boiler and the area of the supply pipe. It is evident that this mode of supply is not convenient in high-pressure boilers; for suppose the pressure to be 50 lbs. per inch, then the height to produce equilibrium will be 115.2 feet. The pressure on an hydraulic ram is frequently 3 tons per inch, in pounds equal to (3 × 2240) 6720 lbs., and 6720 ÷ 0.434 = 15484 feet.

If, instead of a free surface in the cistern, we had supposed a solid piston or plunger to press on the enclosed water, the head should in like manner be calculated by turning the pressure per square inch on the piston into vertical feet of water.

13. Having thus shown the law of the velocities of a fluid issuing from an orifice, let us proceed to apply it to the determination of its discharge, which is defined to be the volume of the fluid which escapes in the unit of time, i. e. one second.

If the mean velocity of all the particles was that due to the "charge" H, then this velocity, which is called the theoretic velocity, would be 2g H; and if at the same time the particles issued from all points of the orifice in parallel threads, it is evident that the volume of water flowing out in a second would be equal to the volume of a prism which would have the orifice for its base, and that velocity for height; and, calling S the area or section of the orifice, the volume of water, or of the prism, would be—

SxV=S√2gH.

This is the theoretic discharge.

14. But the actual discharge is always less than this. In order to have an exact idea of the phenomena, let us consider the fluid vein a short distance after its issue from the orifice, and let us suppose it cut by a plane perpendicular to its direction. It is manifest that the discharge will be equal to the product of the section by the mean velocity of the several threads at the moment they intersect the plane of the section. If this section was equal to that of the orifice, and if this velocity was that due to the charge, then the actual discharge would also be equal to the theoretic discharge. But whether from the section of the vein being considerably less than that of the orifice, as in the flow through orifices in a thin plate, or from the velocity being considerably less than that due to the charge, as in cylindrical adjutages; or, again, from a diminution in both the section and the velocity, as in certain

conical adjutages,-it always results, that the actual discharge is in every case less than the theoretic, and in order to reduce this last to the former, it is necessary to multiply it by some fraction. Let m represent this fraction, and Q the actual discharge, we shall then have

Q = m S√ 2gH,

and designating the volume of water flowing off in the time T by we shall have

Q' = m ST√2gH;

whether the diminution in the discharge arises from a diminution in the section, or in the velocity, it is always a consequence of the contraction which the fluid vein suffers in passing through the orifice, and thus the multiplier m, or "coefficient for the reduction of the theoretic to the actual discharge," is generally called the" coefficient of contraction." Its accurate determination is of the greatest importance; upon its degree of exactness depends that of the results we obtain when we would apply to practice formulæ upon the flow of water. the flow of water. We shall now proceed to give the results of experiments on the value of the symbol m, giving some preliminary statements, upon, first, the cause of the "contraction;" second, upon the nature of its effects; and third, upon the form of the fluid vein-the orifice being circular-its dimensions, and the effect of the form. upon the discharge.

15. Cause of the Contraction. If we take a glass vessel (Fig. 5), in the side of which is an orifice through which the water flows, and render visible the movement of the molecules of the water in the vessel by disseminating through it substances of equal specific gravity, and very minute, or by producing within the water some light chemical precipitation, such as occurs when we let fall a few drops of nitrate of silver in water slightly saline, we then see at a small distance from the orifice as, for instance, about an inch, when its diameter is three-eighths of an inch-the fluid molecules converge from all parts towards the orifice, describing curved lines, and finally, as if precipitated upon a centre of attraction, issue forth with a rapidly increased motion. The convergence of the directions that they had within the vessel at the moment of their arrival at the orifice, still continues for a short distance after they have passed out, so that we can plainly see the fluid vein after its passage gradually diminish, and become contracted up to the place where the particles, from the effect of their

mutual action, and of the motions impressed upon them, take directions, it may be parallel, or in some other lines. The vein thus forms a species of truncated pyramid or cone whose larger base is the orifice, and the smaller is the section of the fluid at its place of greatest contraction, a section which is often called the "section of contraction." This figure, and all the phenomena of contraction, are thus a consequence of the convergence of the several threads of water when they arrive at the orifice, or of the obliquity of their mutual directions.

16. Effects of the Contraction.-When the orifice is in a thin plate, the contraction is completely external to the reservoir; it is thus clearly visible, can be, and in fact it has been, measured, as we shall mention directly. When the orifice is circular, the fluid vein having reached the minimum section, continues of that transverse area, and is thus cylindrical in form, and has a velocity very nearly equal to that due to the charge. The discharge will therefore be the product of this section by the velocity, so that the effect of the contraction is limited to the reduction of the value of the section which enters into the expression of the discharge. The flow will take place as if the actual orifice had been replaced by another whose diameter was equal to the "section of contraction," but in which supposed orifice no true contraction took place.

If to the orifice AB (Fig. 6) we attach a tube or cylindrical adjutage, the fluid threads will arrive at AB, converging, and therefore the fluid will contract at the entrance. Experiments prove that this contraction is identical with that of the thin plate; it will, however, be internal with respect to the mouth of the tube. Moreover, beyond the "section of contraction" the attraction of the sides of the tube occasions a dilatation of the fluid vein; the threads follow these sides, and issue parallel to each other and to the axis of the tube: so that the section of the vein at its exit is fully equal to that of the orifice in the side of the reservoir, but the velocity is not that due to the charge. If the flow was solely produced by the pressure of the fluid, then the velocity at the section of contraction would be that due to the charge, and would diminish in proportion as the form of the fluid vein enlarged by virtue of that law or axiom of hydraulics—namely, when an incompressible fluid is in motion forming a continuous mass-then the velocity, at all its diverse sections, is inversely proportional to the area of the section: the diminution of velocity would then cease