of the cross sections at A and B, that is at inlet and throat, is in actual meters 5 to 1 to 20 to 1, and is very carefully determined by the maker of the meter. Then, if v and u are the velocities at A and B, u = ρv. Let pressure pipes be introduced at A, B and C, and let H1, H, H2 be the pressure heads at those points. Since the velocity at B is greater than at A the pressure will be less. Neglecting friction
Let h = H1 − H be termed the Venturi head, then
from which the velocity through the throat and the discharge of the main can be calculated if the areas at A and B are known and h observed. Thus if the diameters at A and B are 4 and 12 in., the areas are 12.57 and 113.1 sq. in., and ρ = 9,
If the observed Venturi head is 12 ft.,
and the discharge of the main is
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| Fig. 32. |
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| Fig. 33. |
Hence by a simple observation of pressure difference, the flow in the main at any moment can be determined. Notice that the pressure height at C will be the same as at A except for a small loss hf due to friction and eddying between A and B. To get the pressure at the throat very exactly Herschel surrounds it by an annular passage communicating with the throat by several small holes, sometimes formed in vulcanite to prevent corrosion. Though constructed to prevent eddying as much as possible there is some eddy loss. The main effect of this is to cause a loss of head between A and C which may vary from a fraction of a foot to perhaps 5 ft. at the highest velocities at which a meter can be used. The eddying also affects a little the Venturi head h. Consequently an experimental coefficient must be determined for each meter by tank measurement. The range of this coefficient is, however, surprisingly small. If to allow for friction, u = k √ {ρ2/(ρ2 − 1)} √(2gh), then Herschel found values of k from 0.97 to 1.0 for throat velocities varying from 8 to 28 ft. per sec. The meter is extremely convenient. At Staines reservoirs there are two meters of this type on mains 94 in. in diameter. Herschel contrived a recording arrangement which records the variation of flow from hour to hour and also the total flow in any given time. In Great Britain the meter is constructed by G. Kent, who has made improvements in the recording arrangement.
In the Deacon Waste Water Meter (fig. 33) a different principle is used. A disk D, partly counter-balanced by a weight, is suspended in the water flowing through the main in a conical chamber. The unbalanced weight of the disk is supported by the impact of the water. If the discharge of the main increases the disk rises, but as it rises its position in the chamber is such that in consequence of the larger area the velocity is less. It finds, therefore, a new position of equilibrium. A pencil P records on a drum moved by clockwork the position of the disk, and from this the variation of flow is inferred.
§ 33. Pressure, Velocity and Energy in Different Stream Lines.—The equation of Bernoulli gives the variation of pressure and velocity from point to point along a stream line, and shows that the total energy of the flow across any two sections is the same. Two other directions may be defined, one normal to the stream line and in the plane containing its radius of curvature at any point, the other normal to the stream line and the radius of curvature. For the problems most practically useful it will be sufficient to consider the stream lines as parallel to a vertical or horizontal plane. If the motion is in a vertical plane, the action of gravity must be taken into the reckoning; if the motion is in a horizontal plane, the terms expressing variation of elevation of the filament will disappear.[1]
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| Fig. 34. |
Let AB, CD (fig. 34) be two consecutive stream lines, at present assumed to be in a vertical plane, and PQ a normal to these lines making an angle φ with the vertical. Let P, Q be two particles moving along these lines at a distance PQ = ds, and let z be the height of Q above the horizontal plane with reference to which the energy is measured, v its velocity, and p its pressure. Then, if H is the total energy at Q per unit of weight of fluid,
Differentiating, we get
for the increment of energy between Q and P. But
where the last term disappears if the motion is in a horizontal plane.
Now imagine a small cylinder of section ω described round PQ as an axis. This will be in equilibrium under the action of its centrifugal force, its weight and the pressure on its ends. But its volume is ωds and its weight Gω ds. Hence, taking the components of the forces parallel to PQ—
where ρ is the radius of curvature of the stream line at Q. Consequently, introducing these values in (1),
Currents
§ 34. Rectilinear Current.—Suppose the motion is in parallel straight stream lines (fig. 35) in a vertical plane. Then ρ is infinite, and from eq. (2), § 33,
Comparing this with (1) we see that
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| Fig. 35. |
or the pressure varies hydrostatically as in a fluid at rest. For two stream lines in a horizontal plane, z is constant, and therefore p is constant.
Radiating Current.—Suppose water flowing radially between horizontal parallel planes, at a distance apart = δ. Conceive two cylindrical sections of the current at radii r1 and r2, where the velocities are v1 and v2, and the pressures p1 and p2. Since the flow across each cylindrical section of the current is the same,
- ↑ The following theorem is taken from a paper by J. H. Cotterill, “On the Distribution of Energy in a Mass of Fluid in Steady Motion,” Phil. Mag., February 1876.
