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36
[KINEMATICS OF FLUIDS
HYDRAULICS


1 cub. ft.  = 6.236 imp. gallons  = 7.481 U.S. gallons.
1 imp. gallon  = 0.1605 cub. ft.  = 1.200 U.S. gallons.
1 U.S. gallon  = 0.1337 cub. ft.  = 0.8333 imp. gallon.
1 litre  = 0.2201 imp. gallon  = 0.2641 U.S. gallon.

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 p2p1 to the cubical compression is sensibly constant. That is, k = (p2p1)V1/(V2 − V1) is constant. This constant is termed the elasticity of volume. With the notation of the differential calculus,

k = dp / ( dV ) = − V dp .
V dV

Elasticity of Volume of Liquids.

  Canton. Oersted. Colladon
and Sturm.
Regnault.
Water 45,990,000 45,900,000 42,660,000 44,000,000
Sea water 52,900,000 ·· ·· ··
Mercury 705,300,000 ·· 626,100,000 604,500,000
Oil 44,090,000 ·· ·· ··
Alcohol 32,060,000 ·· 23,100,000 ··

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.

Temperature. ρ
Density of
Water.
G
Weight of
1 cub. ft.
in ℔.
 Cent.  Fahr.
0 32.0 .999884 62.417
1 33.8 .999941 62.420
2 35.6 .999982 62.423
3 37.4  1.000004 62.424
4 39.2 1.000013 62.425
5 41.0 1.000003 62.424
6 42.8 .999983 62.423
7 44.6 .999946 62.421
8 46.4 .999899 62.418
9 48.2 .999837 62.414
10 50.0 .999760 62.409
11 51.8 .999668 62.403
12 53.6 .999562 62.397
13 55.4 .999443 62.389
14 57.2 .999312 62.381
15 59.0 .999173 62.373
16 60.8 .999015 62.363
17 62.6 .998854 62.353
18 64.4 .998667 62.341
19 66.2 .998473 62.329
20 68.0 .998272 62.316
22 71.6 .997839 62.289
24 75.2 .997380 62.261
26 78.8 .996879 62.229
28 82.4 .996344 62.196
30 86   .995778 62.161
35 95   .99469  62.093
40 104   .99236  61.947
45 113   .99038  61.823
50 122   .98821  61.688
55 131   .98583  61.540
60 140   .98339  61.387
65 149   .98075  61.222
70 158   .97795  61.048
75 167   .97499  60.863
80 176   .97195  60.674
85 185   .96880  60.477
90 194   .96557  60.275
100  212   .95866  59.844

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

pω = Ghω ℔,
p/G = h;

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.

Relation of Pressure, Temperature, and Density of Gases

EB1911 Hydraulics - Fig. 3.jpg
Fig. 3.

§ 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 p0 be mean atmospheric pressure (2116.8 ℔ per sq. ft.), V0 the volume of 1 ℔ of air at 32° Fahr. under the pressure p0. Then

p0V0 = 26214.
(1)

If G0 is the weight per cubic foot of air in the same conditions,

G0 = 1/V0 = 2116.8/26214 = .08075.
(2)

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

pV = p/G = 26214; or 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,

pV = p0V0 (460.6 + t) / (460.6 + 32) = p0V0τ/τ0,
(4)

where τ, τ0 are the temperatures t and 32° reckoned from the absolute zero, which is −460.6° Fahr.;

p/G = p0τ/G0τ0;
(4a)
G = pτ0G0/p0τ.
(5)

If p0 = 2116.8, G0 = .08075, τ0 = 460.6 + 32 = 492.6, then

p/G = 53.2τ.
(5a)

Or quite generally p/G = Rτ for all gases, if R is a constant varying inversely as the density of the gas at 32° F. For steam 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.