soever N may be, R will always be less than T. This is the principle of the brake applied by Sir William Thomson to apparatus for paying out submarine telegraph cables, with a view to limiting the resistance within the amount which the cable can safely bear.

In any case in which it is desired to give a great value to the ratio N, the flexible brake may be coiled spirally round the drum, so as to make the arc of contact greater than one circumference.

355. Pump-Brakes.-The resistance of a fluid, forced by a pump through a narrow orifice, may be used to dispose of superfluous energy; as in the "cataract," or "dash-pot."

The energy which is expended in forcing a given weight of fluid through an orifice is found by multiplying that weight into the height due to the greatest velocity which its particles acquire in that process, and into a factor greater than unity, which for each kind of orifice is determined experimentally, and whose excess above unity expresses the proportion which the energy expended in overcoming the friction between the fluid and the orifice bears to the energy expended in giving velocity to the fluid.

The following are some of the values of that factor, which will be denoted by 1 + F :—

For an orifice in a thin plate, 1+ F

=== 1.054.

....(1.)

For a straight uniform pipe of the length 7, and whose hydraulic mean depth, that is, the area divided by the circumference of its cross-section, is m,

For cylindrical pipes, m is one-fourth of the diameter.

The factor f in the last formula is called the co-efficient of friction of the fluid. For water in iron pipes, the diameter d being expressed in feet, its value, according to Darcy, is

The greatest velocity of the fluid particles is found by dividing the volume of fluid discharged in a second by the area of the outlet at its most contracted part. When the outlet is a cylindrical pipe, the sectional area of that pipe may be employed in this calculation; but when it is an orifice in a thin plate, there is a contracted vein of the issuing stream after passing the orifice, whose area is on an average about 0-62 of the area of the orifice itself; and that contracted area is to be employed in computing the

velocity. Its ratio to the area of the orifice in the plate is called the co-efficient of contraction. (See page 586.)

The computation of the energy expended in forcing a given quantity of a given fluid in a given time through a given outlet, is expressed symbolically as follows:

Let V be the volume of fluid forced through, in units of volume per second.

D, the heaviness of the fluid (see page 326).

A, the area of the orifice.

R, the resistance overcome by the piston of the pump in driving the water.

the factor 1+ F being computed by means of the formulæ 1, 2, 3, 4. To find the intensity of the pressure (p) within the pump, it is to be observed, as in Article 302, that if A' denotes the area of the piston,

V = A' u; R = p A';....

..(7.)

....(8.)

that is, the intensity of the pressure is that due to the weight of a vertical column of the fluid, whose height is greater than that due to the velocity of outflow in the ratio 1 F: 1.

To allow for the friction of the piston, about one-tenth may in general be added to the result given by equation 6. (Sec page 399.) The piston and pump have been spoken of as single; and such may be the case when the velocity of the piston is uniform. When a piston, however, is driven by a crank on a shaft rotating at an uniform speed, its velocity varies; and when a pump-brake is to be applied to such a shaft, it is necessary, in order to give a sufficiently near approximation to an uniform velocity of outflow, that there should be at least either three single acting pumps, driven by three cranks making with each other angles of 120°, or a pair of double-acting pumps, driven by a pair of cranks at right angles to each other; and the result will be better if the pumps

force the fluid into one common air vessel before it arrives at the resisting orifice.

That orifice may be provided with a valve, by means of which its area can be adjusted so as to cause any required resistance.

A pump-brake of a simple kind is exemplified in the apparatus called the "cataract," for regulating the opening of the steam valve in single-acting steam engines. It is fully described in most special treatises on those engines.*

356. Fan-Brakes-A fan, or wheel with vanes, revolving in water, oil, or air, may be used to dispose of surplus energy; and the resistance which it causes may be rendered to a certain extent adjustable at will, by making the vanes so as to be capable of being set at different angles with their direction of motion, or at different distances from their axis.

Fan-brakes are applied to various machines, and are usually adjusted so as to produce the requisite resistance by trial. It is, indeed, by trial only that a final and exact adjustment can be effected; but trouble and expense may be saved by making, in the first place, an approximate adaptation of the fan to its purpose by calculation.

The following formulæ are the results of the experiments of Duchemin, and are approved of by Poncelet in his Mécanique Industrielle :

:

For a thin flat vane, whose plane traverses its axis of rotation, let A denote the area of the vane;

7, the distance of its centre of area from the axis of rotation; s, the distance from the centre of area of the entire vane to the centre of area of that half of it which lies nearest the axis of rotation;

v, the velocity of the centre of area of the vane (= a l, if a is the angular velocity of rotation);

D, the heaviness of the fluid in which it moves;

Rl, the moment of resistance;

k, a co-efficient whose value is given by the formula

When the vane is oblique to its direction of motion, let i denote

*Pump-brakes have been applied to railway carriages by Mr. Laurence Hill. Hydraulic buffers, which act on the same principle, have been applied to railway carriages by Colonel Clark, R.A.

the acute angle which its surface makes with that direction; then the result of equation 2 is to be multiplied by

It appears that the resistance of a fan with several vanes increases nearly in proportion to the number of vanes, so long as their distances apart are not less at any point than their lengths. Beyond that limit the law is uncertain.

SECTION II-Of Fly-Wheels.

G

K

E

357. Periodical Fluctuations of Speed in a machine (A. M., 689) are caused by the alternate excess and deficiency of the energy exerted above the work performed in overcoming resisting forces, which produce an alternate increase and diminution of actual energy, according to the law explained in Article 330, page 373. To determine the greatest fluctuation of speed in a machine moving periodically, take A B C, in fig. 255, to represent the motion of the driving point during one period; let the effort P of the prime mover at each instant be represented by the ordinate of the curve D G EI F; and let the sum of the resistances, reduced to the driving point as in Article 305, at each instant, be denoted by R, and represented by the ordinate of the curve D HEK F, which cuts the former curve at the ordinates A D, BE, C F. Then the integral,

(PR) d 8,

D

H

B

Fig. 255.

being taken for any part of the motion, gives the excess or deficiency of energy, according as it is positive or negative. For the entire period A B C, this integral is nothing. For A B, it denotes an excess of energy received, represented by the area DGE H; and for B C, an equal excess of work performed, represented by the equal area EK FI. Let those equal quantities be each represented by ΔΕ. Then the actual energy of the machine attains a maximum value at B, and a minimum value at A and C, and ▲ E is the difference of those values.

Now let v be the mean velocity, v1 the greatest velocity, v, the least velocity of the driving point, and En2 W the reduced inertia of the machine (see Article 315, page 362); then

v2 − v2 . Σ • n2 W

which, being divided by the mean actual energy,

and observing that v。 = (v1 + v2) ÷ 2, we find

a ratio which may be called the co-efficient of fluctuation of speed or of unsteadiness.

The ratio of the periodical excess and deficiency of energy ▲ E to the whole energy exerted in one period or revolution, P d 8, has been determined by General Morin for steam engines under 1 various circumstances, and found to be from to for single10 4

1

cylinder engines. For a pair of engines driving the same shaft, with cranks at right angles to each other, the value of this ratio is about one-fourth, and for three engines with cranks at 120', one-twelfth of its value for single-cylinder engines.

The following table of the ratio, a E÷P d s, for one revolution of steam engines of different kinds is extracted and condensed from General Morin's works: