?

water, has been found to be about .615. If we take the value of k =

.64, we shall have,

100

That is, the actual velocity is only 90% of the theoretical velocity. This diminution is due to friction, viscosity, &c.

Efflux through Short Tubes.

177. It is found that the discharge from a given orifice is increased, when the thickness of the plate through which the flow takes place, is increased; also, when a short tube is introduced.

When a tube AB, is employed which is not more than four times as long as the diameter of the orifice, the value of m becomes, on an average, equal to .813; that is, the discharge. per second is 1.325 times greater when the tube is used, than without it. In using the cylindrical tube, the contraction takes place at the outlet of the vessel, and not at the outlet of the tube. Compound mouth-pieces are sometimes used formed of

BD

Fig. 155.

F

B

two conic frustrums, as shown in the figure, having the form of the vein. It has been shown by ETELWEIN, that the most effective tubes of this form should have the diameter of the cross section CD, equal to .833 of the diameter AB. The angle made by the sides CE and DF, should be about 5° 9', and the length of this portion should be three times that of the other.

E

Fig. 156.

EXAMPLES.

1. With what theoretical velocity will water issue from a small orifice 16 feet below the surface of the fluid?

Ans. 324 ft.

2. If the area of the orifice, in the last example, is of a square foot, and the coefficient of efflux .615, how many cubic feet of water will be discharged per minute?

Ans. 118.695 ft.

3. A vessel, constantly filled with water, is 4 feet high, with a cross-section of one square foot; an orifice in the bottom has an area of one square inch. In what time will three-fourths of the water be drawn off, the coefficient of efflux being .6? Ans. 1 minute, nearly.

4. A vessel is kept constantly full of water. How many cubic feet of water will be discharged per minute from an orifice 9 feet below the upper surface, having an area of 1 square inch, the coefficient of efflux being .6 ?

Ans. 6 cubic feet, about.

5. In the last example, what will be the discharge per minute, if we suppose each square foot of the upper surface to be pressed by a force of 645 lbs.?

Ans. 8 cubic feet, about.

6. The head of water in a vessel kept full of water is roo of a square foot. What quantity of water will be discharged per second, when the orifice is through a thin plate?

In this case, we have,

Q

SOLUTION.

= .615 × .01√2 × 321 × 16 = .197 cubic feet.

When a short cylindrical tube is used, we have,

In ETELWEIN'S compound mouthpiece, if we take the smallest cross-section as the orifice, and denote it by a, it is found that the discharge is 24 times that through an orifice of the same size in a thin plate. In this case, we have, supposing a bo of a cubic foot,

=

Motion of water in open channels.

178. When water flows through an open channel, as in a river, canal, or open aqueduct, the form of the channel being always the same, and the supply of water being constant, it is a matter of observation that the flow becomes uniform; that is, the quantity of water that flows through any cross-section, in a given time, is constant. On account of adhesion, friction, &c., the particles of water next the sides and bottom of the channel have their motion retarded. This retardation is imparted to the next layer of particles, but in a less degree, and so on, till a line of particles is reached whose velocity is greater than that of any other filament. This line, or filament of particles, is called the axis of the stream. In the case of cylindrical pipes, the axis coincides sensibly with the axis of the pipe; in straight, open channels, it coincides with that line of the upper surface which is midway between the sides.

A section at right-angles to the axis is called a cross-section, and, from what has been shown, the velocities of the fluid particles will be different at different points of the same cross-section. The mean velocity corresponding to any cross-section, is the average velocity of the particles at every point of that section. The mean velocity may be found by dividing the volume which flows through the section in one second, by the area of the cross-section. Since the same volume flows through each cross-section per second, after the flow has become uniform, it follows that, in channels of varying width, the mean velocity, at any section, will be inversely as the area of the section.

The intersection of the plane of cross-section with the sides and bottom of the channel, is called the perimeter of the section. In the case of a pipe which is constantly filled, the perimeter is the entire line of intersection of the plane of cross-section, with the interior surface of the pipe.

The mean velocity of water in an open channel depends, in the first place, upon its inclination to the horizon. As the inclination becomes greater, the component of gravity in the

direction of the channel increases, and, consequently, the velocity becomes greater. Denoting the inclination by I, and resolving the force of gravity into two components, one at right angles to the upper surface, and the other parallel to it, we shall have for the latter component,

gsin I.

This is the only force that acts to increase the velocity. The velocity will be diminished by friction, adhesion, &c. The total effect of these resistances will depend upon the ratio of the perimeter to the area of the cross section, and also upon the velocity. The cross-section being the same, the resistances will increase as the perimeter increases; consequently, for the same cross-section, the resistance of friction will be the least possible when the perimeter is least possible. The retardation of the flow will also diminish as the area of the cross-section is increased, other things remaining unchanged.

If we denote the area of the cross-section by a, the perimeter, by P, and the velocity, by v, we shall have,

in which ƒ denotes some function of v.

Since the inclination is very small in all practical cases, we may place the inclination itself for the sine of the inclination, and doing so, it has been shown by PRONY, that the function of v may be expressed by two terms, one of which is of the first, and the other of the second degree, with respect to v; or,

in which k and I are constants, to be determined by experiment. According to ETELWEIN, we have,

k = .0000242651, and 7 = .0001114155.

Substituting these values, and solving with respect to v, we have,

ข =

0.1088941604 + √.0118580490 + 8975.414285 RI,

from which the velocity can be found when R and I are known. The values of k and l, and consequently that of v, were found by PRONY to be somewhat different from those given above. Those of ETELWEIN are selected for the reason that they were based upon a much larger number of experiments than those of PRONY.

Having the mean velocity and the area of the cross-section, the quantity of water delivered in any time can be computed. Denoting the quantity delivered in n seconds, by Q, and retaining the preceding notation, we have,

The quantity of water to be delivered is generally one of the data in all practical problems involving the distribution of water. The difference of level of the point of supply and delivery is also known. The preceding principles enable us to give such a form to the cross-section of the canal, or aqueduct, as will ensure the requisite supply.

Were it required to apply the results just deduced, to the case of irregular channels, or to those in which there were many curves, a considerable modification would be required. The theory of these modifications does not come within the limits assigned to this treatise. For a complete discussion of the whole subject of hydraulics in a popular form, the reader is referred to the Traité d'Hydraulique D'AUBISSON.