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In laying pipes the following directions are not unimportant; the mouth, both for ingress and egress, should be trumpetshaped; bends should be as far as possible avoided, and especially sharp angular bends; at junctions the smaller pipe should be brought round in a curve to agree in direction with the main. And, lastly, where a pipe rises and falls much, air is apt to collect in the upper parts of the bends, and thus reduce the section at that part, and it is advisable to make provision by a cock or otherwise for drawing it off at intervals.

Flow of water in open channels.—This is a question of importance, and requires careful consideration on the part of the engineer, as it is a case of frequent occurrence in calculations of the flow of water. It is often, from various circumstances, impossible to throw even a temporary waste board or weir across a stream the quantity of discharge from which it is desirable to ascertain, and hence it becomes necessary to determine a formula which takes into account the friction of the river sides.

In estimating the velocity of a stream on a canal or river, by throwing in floating bodies of nearly the same specific gravity as the water, and estimating the time they require to pass a given measured distance, it must be borne in mind that the velocity is greatest in the centre of the stream and near the surface, and less at the bottom and near the sides. It is generally most convenient to ascertain the velocity at the centre, where the stream is fleetest, but it is essential in calculations to know the mean velocity, or the velocity of a stream of the same section, discharging the same quantity of water, but unaffected by friction at the sides. In practice it will be sufficient to assume that the mean velocity of a stream is equal to 0*83 per cent. or £ of the velocity at the surface.

Or we may use an empirical formula of Prony's, putting v for the mean velocity, and v for the surface velocity, measured by a floating body at the middle of the stream

v + 10-33 v'

For small streams the most accurate method of measurement is the formation of a temporary weir by a vertical board thrown across the stream and carefully puddled at the edges. A rectangular notch of sufficient capacity to pass the water must be cut in the middle portion. The height of the water above the level of the notch should then be measured either at its crest, or better still at some distance behind, where the water is nearly still, and the constant for calculation will be found in Table V. or VI. as the case may be.

But if a waste board or weir cannot be employed, we may find the surface velocity, and from that obtain its mean velocity by the methods given above. If then we take the depth of the stream at various parts of its breadth and so calculate its sectional area, we may find the cubic feet of water discharged per second by multiplying the mean velocity (in feet per second) by the area so found (in square feet).

Thus, if a body floats along the surface of a stream 300 feet

in a minute, its maximum velocity = — 5 feet per second,

60

and its mean velocity, according to Prony, = —

= 4-16. Now let the depth of the stream, 16 feet broad, measured at equal distances of two feet apart, be 0, 1, 2^, 3, 3, 3f, 3, If, 0 feet respectively, then the area =

2 x (1 + 21 + 3 + 3 + 3f + 3 + If) = 36 sq. ft. .-. Cubic feet of water discharged per second

= 36 x 4-16 = 149-76 cubic feet.

In rivers, the coefficient of friction n in formulae (12) (13) and (14) may be taken at 0*0075; it varies, according to Weisbach, from 0-00811 to 0-00748 as the velocity increases from 0-1 to 1-0 metres, and from 0-00748 to 0-00743 as the velocity increases from 1 to 3 metres.

The following formulae express the relations of velocity, fall, and discharge, when the flow of the stream is uniform:

i7•••(*»

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Where h = whole fall, Q = discharge, F = transverse section of stream, c = mean velocity of stream, I = distance which

the river flows for a fall h,^ the perimeter of the water profile,

and f the coefficient of friction.

The best form of section must be that which presents the least resistance to a given quantity of water flowing through the channel. Now it has been shown that the resistance of friction varies directly as the wetted perimeter and inversely as the area of the section, and when the area is constant it will therefore vary directly as the wetted perimeter. Consequently the best form of section will be that with the least perimeter for a given area. Hence for open channels in which the upper water line is not part of the wetted perimeter, the half square is the best rectangular section, and the semihexagon the best trapezoidal section. For equal flows of water the semicircle will have less friction than the semihexagon, and the semihexagon than the semisquare. In designing conduits, for instance, the head race or tail race of water wheels, not only must the sectional form be attended to, but bends must be avoided as much as possible.

Estimation of Water Power.—Where a natural reservoir of mechanical power is employed through the medium of a primemover in overcoming resistances, in sawing, grinding, &c., we term the moving force the power, and the resistances overcome the work.

The dynamic unit by which we estimate force or resistance is the foot-pound, or the unit of force which is capable of lifting a weight of one pound one foot high. A second unit is employed when estimating large expenditure of force, namely, the horse-power. One horse-power, according to the estimate of Watt, was equivalent to 33,000 lbs. raised one foot high in a minute or 550 foot-pounds per second.

It is evident that a power exerted by a weight of water falling a given number of feet is capable of raising an equal weight the same number of feet. The power expended must equal the resistances overcome. In transmitting power through a prime-mover, however, a certain loss necessarily takes place, arising (1) from the loss or waste of the power by spilling, leakage, &c, and (2) from the absorption of a part of the power in overcoming the resistances of the prime-mover itself, friction, &c. Hence the work accomplished by a prime-mover is never equivalent to the power expended on it; the useful effect is always only a certain percentage of the power, and this percentage is called the efficiency or modulus of the machine.

Now for a water-wheel on which a stream of water acts by gravity alone:—

Let h = height of fall in feet.

w = weight of water delivered on the wheel per second.
n = the number of cubic feet per second.
P = the dynamic force of the falling water in foot-pounds,
p, = p reduced to horses power.

c = the useful effect of the machine in foot-pounds and 11,= v reduced to H. P. u = the modulus of the machine. Then for the total water power of the wall we have, in footpounds,

p = w h ... (1); and water weighing 62*5 lbs. per cubic foot, p = 62-5 n h;

or, in horses power,

_ w h _ 62*5 nh _nh P' ~~ 550 550 ^8

Hence, for every foot of fall 8*88 cubic feet of water per second, or 1*47 tons per minute, theoretically afford an available force of one horse power.

But by definition,

155-"...(*>

._•. u in foot-pounds.

r\Jb w h

.-. u,= in horses power.

1 55000 F

and,

100—u _ ^ gum q£ ^ resistances from friction, &c., 100

and the loss from wasted water, in accumulating and transmitting the power.

PART I. I

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