« AnteriorContinuar »
What pressure will a board sustain, placed diagonally through a vessel, the side of which is 9 feet deep, and bottom 12 feet by 9 feet? ✓ 122 +92=15 feet, the length of diagonal board. 15 x 9 x 41 x 1000
=37969 lbs. nearly. 16 Though the diagonal board bisects the vessel, yet it sustains more than half of the pressure in the bottom, for the area of bottom is 12 x 9, and the half of the pressure is : of 60750=30375.
The bottom of a conical or pyramidical vessel sustains a pressure equal to the area of the bottom and depth of water, consequently, the excess of pressure is three times the weight of water in the vessel.
WATER (Hydraulics.) Hydraulics is that science which treats of fluids considered as in motion; it therefore embraces the phenomena exhibited by water issuing from orifices in reservoirs, projected obliquely, or perpendicularly, in jets-d'eau, moving in pipes, canals, and rivers, oscillating in waves, or opposing a resistance to the progress of solid bodies.
It would be needless here to go into the minutiæ of hydraulics, particularly when the theory and practice do not agree. It is only the general laws, deduced from experiment, that can be safely employed in the various operations of hydraulic architecture.
Mr. Banks, in his Treatise on Mills, after enumerating a number of experiments on the velocity of flowing water, by several philosophers, as well as his own, takes from thence the following simple rule, which is as near the truth as any that have been stated by other experimentalists.
RULE. --Measure the depth of the vessel, &c.) in feet, extract the square root of that depth, and multiply it by 5.4, which gives the velocity in feet per second; this multiplied by the area of the orifice in feet, gives the number of cubic feet which flows out in one second.
EXAMPLE. Let a sluice be 10 feet below the surface of the water, its length 4 feet, and open 7 inches ; required the quantity of water expended in one second ?
✓ 1033.162 5.4=17.0748 feet velocity. 4x7
=2} feet x 17.0748=39.84 cubic feet of water per 12 second.
If the area of the orifice is great compared with the head, take the medium depth, and two-thirds of the velocity from that depth, for the velocity.
Given the perpendicular depth of the orifice 2 feet, its borizontal length 4 feet, and its top 1 foot below the surface of water. To find the quantity discharged in one second:
The medium depth is =1.5 X 5.1=8.10 3 of 8.10=5.40, and 5.40 x8=43:20 cubic feet. *
The quantity of water discharged through slits, or notches, 'cut in the side of a vessel or dam, and open at the top, will be found by multiplying the velocity at the bottom by the depth, and taking ; of the product of the area; which again multiplied by the breadth of the slit or notch, gives the quantity of cubic feet discharged in a given time.
EXAMPLE. Let the depth be 5 inches, and the breadth 6 inches; required the quantity run out in 46 seconds ?
The depth is .4166 of a foot.
The breadth is .5 of a foot. ✓ .4166=.6445 x 5.4 x 3=2.3238 x.4166 =.96825 x .5=.48412 feet per second.
Then .48412 x 46=222.69 cubic feet in 46 seconds.
There are two kinds of water wheels, Undershot and Overshot. Undershot, when the water strikes the wheel at, or below the centre. Oversbot when the water falls upon the wheel above the centre.
* The square root of the depth is not taken in this example, but when the depth is considerable, it ought to be taken,
The effect produced by an undershot wheel, is from the impetus of the water. The effect produced by an overshot wheel, is from the gravity or weight of the water.
Of an undershot wheel, the power is to the effect as 3:1. Of an overshot wheel, the power is to the effect as 3:2 which is double the effect of an undershot wheel.
The velocity at a maximum is=3 feet in one second.
Since the effect of the overshot is double that of the undershot, it follows that the higher the wheel is in proportion to the whole descent, the greater will be the effect.
The maximum load for an overshot wheel is that which reduces the circumference of the wheel to its proper velocity,=3 fect in one second; and this will be known by dividing the effect it ought to produce in a given time, by the space intended to be described by the circumference of the wheel in the same time; the quotient will be the resistance overcome at the circumference of the wheel, and is equal to the load required, the friction and resistance of the machinery included.
The following is an extract from Banks on Mills. The effect produced by a given stream in falling through a given space, if compared with a weight, will be directly as that space; but if we measure it by the velocity communicated to the wheel, it will be as the square root of the space descended through, agreeably to the laws of falling bodies.
Experiment 1. A given stream is applied to a wheel at the centre; the revolutions
minute are 38.5. Ex. 2. The same stream applied at the top, turns the same wheel 57 times in a minute.
If in the first experiment the fall is called 1, in the second it will be 2; then the v1:72 :: 38.5 : 54.4, which are in the same ratio as the square roots of the spaces fallen through, and near the observed velocity.
In the following experiments a fly is connected with the water wheel.
Ex. 3. The water is applied at the centre, the wheel revolves 13.03 times in one minute.
E.c. 4. The water is applied at the vertex of the wheel, and it revolves 18.2 times per minute.
As 13.03 : 18.2 :: 71:72 nearly. From the above we infer, that the circumferences of wheels of different sizes may move with velocities which are as the square roots of their diameters without disadvantage, compared one with another, the water in all being applied at the top of the wheel, for the velocity of falling water at the bottom or end of the fall is as the time, or as the square root of the space fallen through; for example, let the fall be 4 feet, then, As ✓16:1" :: 4:}", the time of falling through 4 feet :-Again, let the fall be 9 feet, then, ✓16: 1" :: ✓9:4", and so for
other space, as in the following table, where it appears that water will fall through one foot in a quarter of a second, through 4 feet in half a second, through 9 feet in 3 quarters of a second, and through 16 feet in one second. And if a wheel 4 feet in diameter moved as fast as the water, it could not revolve in less than 1.5 second, neither could a wheel of 16 feet diameter revolve in less than three seconds; but though it is impossible for a wheel to move as fast as the stream which turns it, yet, if their velocities bear the same ratio to the time of the fall through their diameters, the wheel 16 feet in diameter may move twice as fast as the wheel 4 feet in diameter.
Height Time of of the fall falling in in feet. seconds.
Height Time of of the fall falling in in feet. seconds.
5 6 7 8 9 10 12
.25 .352 .432 :.5 .557 .612 .666 .706 .75 .79 .864
14 16 2C 24 25 30 36 40 45 50
.935 1. 1.117 1.22 1.25 1.37 1.5 1.58 1.67 1.76.
POWER AND EFFECT.—The power water has to produce mechanical effect, is as the quantity and fall of perpendicular height. The mechanical effect of a wheel is as the quantity of water in the buckets and the velocity.
The power is to the effect as 3 : 2, that is, suppose the power to be 9000, the effect will be 9000 x 2 18000
3 HEIGHT OF THE WHEEL. The higher the wheel is in proportion to the fall, the greater will be the effect, because it depends less upon the impulse, and more upon the gravity of the water; however, the head should be such, that the water will have a greater velocity than the circumference of the wheel; and the velocity that the circumference of the wheel ought to have, being known, the head required to give the water its proper velocity, can easily be known from the rules of Hydrostatics.
VELOCITY OF THE WHEEL. — Banks, in the foregoing quotation, says that the cireumferences of overshot wheels of different sizes may move with velocities as the square roots of their diameters, without disadvantage. Smeaton says, Experience confirms that the velocity of 3 feet per second is applicable to the highest overshot wheels, as well as the lowest; though high wheels may deviate further from this rule, before they will lose their power, by a given aliquot part of the whole, than low ones can be admitted to do; for a 24 feet wheel may move at the rate of 6 feet per second, without losing any considerable part of its power.
It is evident that the velocities of wheels will be in proportion to the quantity of water and the resistance to be overcome :-if the water flows slowly upon the wheel, more time is required to fill the buckets than if the water flowed rapidly; and whether Smeaton or Banks is taken as a data, the millwright can easily calculate the size of bis wheel, when the velocity and quantity of water in a given time is known.
EXAMPLE I. What power is a stream of water equal to, of the fol. lowing dimensions, viz. 12 inches deep, 22 inches broad;