Page:EB1911 - Volume 14.djvu/48

Rh

Density of Water.—Water at 53° F. and ordinary pressure contains 62·4 ℔ per cub. ft., or 10 ℔ per imperial gallon at 62° F. The litre contains one kilogram of water at 4° C. or 1000 kilograms per cubic metre. River and spring water is not sensibly denser than pure water. But average sea water weighs 64 ℔ per cub. ft. at 53° F. The weight of water per cubic unit will be denoted by G. Ice free from air weighs 57·28 ℔ per cub. ft. (Leduc).

§ 6. Compressibility of Liquids.—The most accurate experiments show that liquids are sensibly compressed by very great pressures, and that up to a pressure of 65 atmospheres, or about 1000 ℔ per sq. in., the compression is proportional to the pressure. The chief results of experiment are given in the following table. Let V1 be the volume of a liquid in cubic feet under a pressure p1 ℔ per sq. ft., and V2 its volume under a pressure p2. Then the cubical compression is (V2 − V1)/V1, and the ratio of the increase of pressure p2 − p1 to the cubical compression is sensibly constant. That is, k = (p2 − p1)V1/(V2 − V1) is constant. This constant is termed the elasticity of volume. With the notation of the differential calculus,

According to the experiments of Grassi, the compressibility of water diminishes as the temperature increases, while that of ether, alcohol and chloroform is increased.

§ 7. Change of Volume and Density of Water with Change of Temperature.—Although the change of volume of water with change of temperature is so small that it may generally be neglected in ordinary hydraulic calculations, yet it should be noted that there is a change of volume which should be allowed for in very exact calculations. The values of ρ in the following short table, which gives data enough for hydraulic purposes, are taken from Professor Everett’s System of Units.

Density of Water at Different Temperatures.

The weight per cubic foot has been calculated from the values of ρ, on the assumption that 1 cub. ft. of water at 39·2° Fahr. is 62·425 ℔. For ordinary calculations in hydraulics, the density of water (which will in future be designated by the symbol G) will be taken at 62·4 ℔ per cub. ft., which is its density at 53° Fahr. It may be noted also that ice at 32° Fahr. contains 57·3 ℔ per cub. ft. The values of ρ are the densities in grammes per cubic centimetre.

§ 8. ''Pressure Column. Free Surface Level''.—Suppose a small vertical pipe introduced into a liquid at any point P (fig. 3). Then the liquid will rise in the pipe to a level OO, such that the pressure due to the column in the pipe exactly balances the pressure on its mouth. If the fluid is in motion the mouth of the pipe must be supposed accurately parallel to the direction of motion, or the impact of the liquid at the mouth of the pipe will have an influence on the height of the column. If this condition is complied with, the height h of the column is a measure of the pressure at the point P. Let ω be the area of section of the pipe, h the height of the pressure column, p the intensity of pressure at P; then

that is, h is the height due to the pressure at p. The level OO will be termed the free surface level corresponding to the pressure at P.

§ 9. Relation of Pressure, Volume, Temperature and Density in Compressible Fluids.—Certain problems on the flow of air and steam are so similar to those relating to the flow of water that they are conveniently treated together. It is necessary, therefore, to state as briefly as possible the properties of compressible fluids so far as knowledge of them is requisite in the solution of these problems. Air may be taken as a type of these fluids, and the numerical data here given will relate to air.

Relation of Pressure and Volume at Constant Temperature.—At constant temperature the product of the pressure p and volume V of a given quantity of air is a constant (Boyle’s law).

Let $$p_0$$ be mean atmospheric pressure (2116·8 ℔ per sq. ft.), $$\mathrm{V}_0$$ the volume of 1 ℔ of air at 32° Fahr. under the pressure $$p_0$$. Then

$$p_0\mathrm{V}_0 = 26214$$. (1)

If $$G_0$$ is the weight per cubic foot of air in the same conditions,

$$\mathrm{G}_0 = 1 / \mathrm{V}_0 = 2116.8 / 26214 = .0807$$. (2)

For any other pressure $$p$$, at which the volume of 1 ℔ is $$\mathrm{V}$$ and the weight per cubic foot is $$\mathrm{G}$$, the temperature being 32° Fahr.,

$$p\mathrm{V} = p/\mathrm{G} = 26214$$; or $$\mathrm{G} = p/26214$$. (3)

Change of Pressure or Volume by Change of Temperature.—Let p0, V0, G0, as before be the pressure, the volume of a pound in cubic feet, and the weight of a cubic foot in pounds, at 32° Fahr. Let p, V, G be the same quantities at a temperature t (measured strictly by the air thermometer, the degrees of which differ a little from those of a mercurial thermometer). Then, by experiment,

$$p\mathrm{V} = p_0 \mathrm{V}_0 (460.6 + t) / (460.6 + 32) = p_0 \mathrm{V}_0 \tau/\tau_0$$, (4)

where $$\tau, \tau_0$$ are the temperatures t and 32° reckoned from the absolute zero, which is −460·6° Fahr.;

$$p/\mathrm{G} = p_0 \tau/\mathrm{G}_0 \tau_0$$; (4a)

$$\mathrm{G} = p\tau_0 \mathrm{G}_0/p_0\tau$$. (5)

If $$\tau_0 = 2116.8, \mathrm{G}_0= .08075, \tau_0 = 460.6 + 32 = 492.6$$, then

$$p/ \mathrm{G} = 53.2\tau $$. (5a)

Or quite generally $$p/\mathrm{G} = \mathrm{R}\tau$$ for all gases, if $$\mathrm{R}$$ is a constant varying inversely as the density of the gas at 32° F. For steam $$\mathrm{R}$$ = 85·5.

II. KINEMATICS OF FLUIDS

§ 10. Moving fluids as commonly observed are conveniently classified thus:

(1) Streams are moving masses of indefinite length, completely or incompletely bounded laterally by solid boundaries. When the solid boundaries are complete, the flow is said to take place in a pipe. When the solid boundary is incomplete and leaves the upper surface of the fluid free, it is termed a stream bed or channel or canal.

(2) A stream bounded laterally by differently moving fluid of the same kind is termed a current.

(3) A jet is a stream bounded by fluid of a different kind.

(4) An eddy, vortex or whirlpool is a mass of fluid the particles of which are moving circularly or spirally.

(5) In a stream we may often regard the particles as flowing along definite paths in space. A chain of particles following each other along such a constant path may be termed a fluid filament or elementary stream.

§ 11. Steady and Unsteady, Uniform and Varying, Motion.—There are two quite distinct ways of treating hydrodynamical questions. We may either fix attention on a given mass of fluid and consider its changes of position and energy under the action of the stresses to which it is subjected, or we may have regard to a given fixed portion of space, and consider the volume and energy of the fluid entering and leaving that space.