the expansion being adiabatic, because in the flow of the streams of air through an orifice no sensible amount of heat can be communicated from outside.
Suppose the air flows from a vessel, where the pressure is p1 and the velocity sensibly zero, through an orifice, into a space where the pressure is p2. Let v2 be the velocity of the jet at a point where the convergence of the streams has ceased, so that the pressure in the jet is also p2. As air is light, the work of gravity will be small compared with that of the pressures and expansion, so that z1z2 may be neglected. Putting these values in the equation above—
| v22/2g = (p1/G1) {γ/(γ − 1)} {1 − (p2/p1)(γ−1)/γ}; | (1) |
an equation commonly ascribed to L. J. Weisbach (Civilingenieur, 1856), though it appears to have been given earlier by A. J. C. Barre de Saint Venant and L. Wantzel.
It has already (§ 9, eq. 4a) been seen that
where for air p0 = 2116.8, G0 = .08075 and τ0 = 492.6.
| v22/2g = {p0τ1γ / G0τ0 (γ − 1)} {1 − (p2/p1)(γ−1)/γ}; | (2) |
or, inserting numerical values,
| v22/2g = 183.6τ1 {1 − (p2/p1)0.29}; | (2a) |
which gives the velocity of discharge v2 in terms of the pressure and absolute temperature, p1, τ1, in the vessel from which the air flows, and the pressure p2 in the vessel into which it flows.
Proceeding now as for liquids, and putting ω for the area of the orifice and c for the coefficient of discharge, the volume of air discharged per second at the pressure p2 and temperature τ2 is
| = 108.7cω √ [τ1 {1 − (p2/p1)0.29}]. | (3) |
If the volume discharged is measured at the pressure p1 and absolute temperature τ1 in the vessel from which the air flows, let Q1 be that volume; then
| = 108.7cω √ (τ1ψ). | (4) |
The weight of air at pressure p1 and temperature τ1 is
Hence the weight of air discharged is
| = 2.043cωp1 √ (ψ/τ1). | (5) |
Weisbach found the following values of the coefficient of discharge c:—
| Conoidal mouthpieces of the form of the | ||||
| contracted vein with effective pressures | c = | |||
| of .23 to 1.1 atmosphere | 0.97 | to | 0.99 | |
| Circular sharp-edged orifices | 0.563 | ” | 0.788 | |
| Short cylindrical mouthpieces | 0.81 | ” | 0.84 | |
| The same rounded at the inner end | 0.92 | ” | 0.93 | |
| Conical converging mouthpieces | 0.90 | ” | 0.99 | |
§ 64. Limit to the Application of the above Formulae.—In the formulae above it is assumed that the fluid issuing from the orifice expands from the pressure p1 to the pressure p2, while passing from the vessel to the section of the jet considered in estimating the area ω. Hence p2 is strictly the pressure in the jet at the plane of the external orifice in the case of mouthpieces, or at the plane of the contracted section in the case of simple orifices. Till recently it was tacitly assumed that this pressure p2 was identical with the general pressure external to the orifice. R. D. Napier first discovered that, when the ratio p2/p1 exceeded a value which does not greatly differ from 0.5, this was no longer true. In that case the expansion of the fluid down to the external pressure is not completed at the time it reaches the plane of the contracted section, and the pressure there is greater than the general external pressure; or, what amounts to the same thing, the section of the jet where the expansion is completed is a section which is greater than the area ccω of the contracted section of the jet, and may be greater than the area ω of the orifice. Napier made experiments with steam which showed that, so long as p2/p1 > 0.5, the formulae above were trustworthy, when p2 was taken to be the general external pressure, but that, if p2/p1 < 0.5, then the pressure at the contracted section was independent of the external pressure and equal to 0.5p1. Hence in such cases the constant value 0.5 should be substituted in the formulae for the ratio of the internal and external pressures p2/p1.
It is easily deduced from Weisbach’s theory that, if the pressure external to an orifice is gradually diminished, the weight of air discharged per second increases to a maximum for a value of the ratio
| p2/p1 = {2/(γ + 1)}γ−1/γ |
For a further decrease of external pressure the discharge diminishes,—a result no doubt improbable. The new view of Weisbach’s formula is that from the point where the maximum is reached, or not greatly differing from it, the pressure at the contracted section ceases to diminish.
A. F. Fliegner showed (Civilingenieur xx., 1874) that for air flowing from well-rounded mouthpieces there is no discontinuity of the law of flow, as Napier’s hypothesis implies, but the curve of flow bends so sharply that Napier’s rule may be taken to be a good approximation to the true law. The limiting value of the ratio p2/p1, for which Weisbach’s formula, as originally understood, ceases to apply, is for air 0.5767; and this is the number to be substituted for p2/p1 in the formulae when p2/p1 falls below that value. For later researches on the flow of air, reference may be made to G. A. Zeuner’s paper (Civilingenieur, 1871), and Fliegner’s papers (ibid., 1877, 1878).
VII. FRICTION OF LIQUIDS.
§ 65. When a stream of fluid flows over a solid surface, or conversely when a solid moves in still fluid, a resistance to the motion is generated, commonly termed fluid friction. It is due to the viscosity of the fluid, but generally the laws of fluid friction are very different from those of simple viscous resistance. It would appear that at all speeds, except the slowest, rotating eddies are formed by the roughness of the solid surface, or by abrupt changes of velocity distributed throughout the fluid; and the energy expended in producing these eddying motions is gradually lost in overcoming the viscosity of the fluid in regions more or less distant from that where they are first produced.
The laws of fluid friction are generally stated thus:—
1. The frictional resistance is independent of the pressure between the fluid and the solid against which it flows. This may be verified by a simple direct experiment. C. H. Coulomb, for instance, oscillated a disk under water, first with atmospheric pressure acting on the water surface, afterwards with the atmospheric pressure removed. No difference in the rate of decrease of the oscillations was observed. The chief proof that the friction is independent of the pressure is that no difference of resistance has been observed in water mains and in other cases, where water flows over solid surfaces under widely different pressures.
2. The frictional resistance of large surfaces is proportional to the area of the surface.
3. At low velocities of not more than 1 in. per second for water, the frictional resistance increases directly as the relative velocity of the fluid and the surface against which it flows. At velocities of 12 ft. per second and greater velocities, the frictional resistance is more nearly proportional to the square of the relative velocity.
In many treatises on hydraulics it is stated that the frictional resistance is independent of the nature of the solid surface. The explanation of this was supposed to be that a film of fluid remained attached to the solid surface, the resistance being generated between this fluid layer and layers more distant from the surface. At extremely low velocities the solid surface does not seem to have much influence on the friction. In Coulomb’s experiments a metal surface covered with tallow, and oscillated in water, had exactly the same resistance as a clean metal surface, and when sand was scattered over the tallow the resistance was only very slightly increased. The earlier calculations of the resistance of water at higher velocities in iron and wood pipes and earthen channels seemed to give a similar result. These, however, were erroneous, and it is now well understood that differences of roughness of the solid surface very greatly influence the friction, at such velocities as are common in engineering practice. H. P. G. Darcy’s experiments, for instance, showed that in old and incrusted water mains the resistance was twice or sometimes thrice as great as in new and clean mains.
§ 66. Ordinary Expressions for Fluid Friction at Velocities not Extremely Small.—Let f be the frictional resistance estimated in pounds per square foot of surface at a velocity of 1 ft. per second; ω the area of the surface in square feet; and v its velocity in feet per second relatively to the water in which it is immersed. Then, in accordance with the laws stated above, the total resistance of the surface is
| R = fωv2 | (1) |
where f is a quantity approximately constant for any given surface. If
| R = ξGωv2/2g, | (2) |
where ξ is, like f, nearly constant for a given surface, and is termed the coefficient of friction.
The following are average values of the coefficient of friction for water, obtained from experiments on large plane surfaces, moved in an indefinitely large mass of water.
