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The quantity A÷s is called the hydraulic mean depth. In Hydraulic a pipe of square or circular section running full, this is evidently mean depth. one-fourth of the diameter. It is also one-fourth of the diameter in a semi-cylindrical open conduit running full, and in one whose sides are tangents to a semicircle, and greatest depth of water one-half of the diameter of that semicircle. The square and circle in close pipes, and the semicircle and the halves of regular polygons in open conduits, are the figures of cross section of least resistance.
(2.) Let denote the angle through which a pipe is bent; Factor of then at a knee, the factor of the work done in changing the friction at
When the pipe makes a circular sweep instead of a sharp turn, let denote two right angles, d the diameter of the pipe, and p the radius of the sweep: then
knees and bends.
At a sharp turn of 90° in a pipe F o'985. To divert the stream through the same angle with a bend of a radius five times the diameter of the pipe, according to equation (14), F = 0·066. In an open channel the velocity of the stream is diminished and its depth increased at and near a knee or bend. In a close pipe the effect of a bend or knee is to diminish the velocity throughout.
(3.) When the velocity of a stream is diminished by a sudden Factor of enlargement of the sectional area of the conduit, it is found that friction at all the energy due to the difference of the velocities in the largements. smaller and larger parts is expended in forming eddies in the water. Let v be the greater velocity, v the less velocity, and let v÷v=r: then at a sudden enlargement—
When the sluice of a stop-cock in a close pipe is partially closed, the effective area below the sluice is considerably less than the real area. For instance, if the real area below the sluice is one-half of the area of the pipe, the effective area is only about one-third of the area of the pipe. The velocity below the sluice in this case, therefore, is three times the velocity on either side, and F = (3—1)2.
Straight uni- In uniform conduits free from bends sharp enough to cause form conduits. appreciable resistance, all the relations between the loss of head, the volume of flow, and the dimensions of the conduit, can be deduced from the following equations:
Equation (17) is derived from equations (7) and (10). The quantity h÷l is called the declivity of the stream.
The following approximate formulæ, applicable to iron pipes, are derived from equations (16) and (17) by substituting o'00645 for f, 64'4 for 2g, 3'1416 d for s, and 0.7854 d2 for A :
The following, applicable to open conduits, are derived by substituting 000756 for f:
Measurement of flow in streams.
These formulæ are exact enough for practical purposes.
The discharge of a stream may be found in three ways: (1.) By measuring the sectional area, finding the mean velocity directly by means of an instrument called a current meter, and by substituting in equation (16).
(2.) By measuring the sectional area, the wetted girth, and the declivity in a part of the stream where the channel is nearly uniform, and by substituting in equation (21).
(3.) By damming the stream so as to make the velocity behind the obstruction imperceptible, measuring the head of pressure under which the water escapes through an orifice of known figure and area, and by substituting in the equations. given in the sequel.
A dam or obstruction in the bed of a stream, when made for the purpose of measuring the flow, is called a weir gauge. The orifice through which the stream escapes is usually a rectangular notch, in a vertical board forming the top of the weir. Sometimes the orifice is a hole, circular or rectangular, in a vertical plate some depth below the water-level.
In a jet escaping at a sharp-edged orifice, the frictional sur- Discharge at face is practically nothing. When the water issues from a still sharp-edged pond, therefore, the loss of head
Let b denote the width of a sharp-edged rectangular orifice whose sides are vertical. Conceive the water issuing from it to be made up of a number of horizontal layers, each of the small thickness dh: then, if b be the width, the sectional area of each of these layers is
and the volume of flow of each
Let h, denote the depth of the lower horizontal edge of the
But the discharges at sharp-edged orifices, when intercepted Coefficients of and measured in a vessel, are found to be less than Q'. The contraction. ratio of the actual to the theoretical discharge is called the coefficient of contraction. Let denote this coefficient, and Q and the actual and theoretical discharges: then
At a notch, h。 = o, and when the notch is sharp-edged and Discharge at rectangular, and has its sides vertical
When the width of the notch is one-fourth of the width of the weir, is found to be o'595; therefore substituting
At a notch the whole width of the weir c=0·667, and
Discharge at flat or roundcrested weir.
Discharge at sharp-edged circular hole.
Gross energy of mill race.
Effective energy of mill race.
At a weir with a flat or round crest c = 0.5 nearly, and
At a sharp-edged circular hole, when 3 (h-ho) is not greater than ho, let h denote the depth below still water of the centre of the hole, and A the area of the hole: then the following is exact enough :
A portion of the energy expended by a stream may be saved by diminishing the resistance of the channel, and converted by means of engines, such as water-wheels, into work done under control. The resistance of the channel is diminished by erecting a weir across it so as to increase the depth and diminish the velocity of the stream for a certain distance. Sometimes the whole or a portion of the water is diverted from the natural channel for a certain distance in an artificial channel. A given length of a stream in which the water-level may be altered, or from which the whole or a portion of the water may be diverted, constitutes a mill race.
Let x be the height of the upper surface of the stream at the beginning, and x, the height of the upper surface of the stream at the end of the mill race; also let D be the weight of a cubic foot of water (62'4 lbs. nearly): then
DQ (x − x1).
is the gross energy of the race in foot pounds per second. Of this gross energy a portion is wasted in the altered channel before the water reaches the engine, a portion in the engine itself, and a portion in returning the water to the stream after leaving the engine. In extreme cases the energy transmitted by the engine varies from one-third to three-fourths of the gross Horse power. energy. The energy in foot pounds per second, divided by 550, is called the horse power of the race or of the engine, as the case may be.
§ 3. ON WATER-WORKS AND DRAINAGE-Works.
The total annual supply of rain to a catchment basin is found by multiplying the area of the basin into the annual mean depth of rainfall. The annual depth of rainfall varies in different districts. It also varies in the same district in different years. At Greenwich the average annual depth is about 25 inches; on the Cumberland hills it is about 140 inches.
The annual yield of a catchment basin, or the water annually Available carried away by its streams above and below ground, is the rainfall or difference between the total annual rain supply and the annual yield. evaporation and permanent absorption.
The absorption of the energy of flowing water by the sur- Natural faces with which it comes in contact causes a storage in soils, storage. rocks, and channels in times of flood, and so maintains a flow
in streams in times of drought.
Works for the purpose of controlling the flow of water are Water-works called water-works. A water-works may have for its object the defined. supply of a constant daily demand for water for domestic and town purposes, for power or for manufactures, the supply of water for irrigation or for canal navigation, the prevention of periodical inundations of flat districts, or a combination of two or more of these objects.
The object of drainage-works is to facilitate the escape of Drainageworks defined.
A water-works consists essentially of a system of channels Essential conveying water from a gathering-ground; a reservoir in which parts of a the water is stored; a sluice, and a system of channels leading to the places where the supply is wanted. It may be either partly or wholly artificial. When a town or district is supplied from a river by pumping or by diversion cuts, the river serves as a natural reservoir. A great mass of porous rock may also act as a natural reservoir when the supply is derived from wells or springs. When the house-top is the gathering-ground, the water-works may be said to be wholly artificial.
for water in towns.
The demand of a town for water depends upon the habits Estimation of and the number of the inhabitants. In general a plentiful the demand supply of water tends to increase the quantity of water used. A safe average to assume in designing town water-works is a daily demand per head of 15 gallons for domestic purposes, and 10 gallons per head for trade and public purposes in what may be called non-manufacturing towns. In manufacturing towns the total demand per head may be taken at 10 gallons more. Allowance must also be made for loss by leakage. The water wasted in this way through neglect not unfrequently amounts to three times the quantity used.
The channels on the gathering-ground may be simply the Drainage of natural water-courses of the district, or those with additions or gatheringground. improvements. The more rapid the conveyance of the water to the deeper channels, the less the loss by evaporation and the greater the yield. As the purity of the water is of importance in the case of a town supply, it may be necessary to divert certain streams. To intercept and raise water flowing underground, wells must Interception be sunk to such a depth that the pumps worked in them may and raising of underground reach below the line of actual or virtual declivity of the currents. As the position of this line of declivity varies with the rainfall, the pumps must reach below its lowest position to give