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 V_{1} be
the volume of a liquid in cubic feet under a pressure *p*_{1} ℔ per sq. ft.,
and V_{2} its volume under a pressure *p*_{2}. Then the cubical compression
is (V_{2} − V_{1})/V_{1}, and the ratio of the increase of pressure
*p*_{2} − *p*_{1} to the cubical compression is sensibly constant. That is,
*k* = (*p*_{2} − *p*_{1})V_{1}/(V_{2} − V_{1}) 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

*p*ω = G

*h*ω ℔,

*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

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 *p*_{0} be mean atmospheric pressure (2116.8 ℔ per sq. ft.), V_{0}
the volume of 1 ℔ of air at 32° Fahr. under the pressure *p*_{0}. Then

*p*

_{0}V

_{0}= 26214.

If G_{0} is the weight per cubic foot of air in the same conditions,

_{0}= 1/V

_{0}= 2116.8/26214 = .08075.

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.,

*p*V =

*p*/G = 26214; or G =

*p*/26214.

*Change of Pressure or Volume by Change of Temperature.*—Let *p*_{0},
V_{0}, G_{0}, 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*V =

*p*

_{0}V

_{0}(460.6 +

*t*) / (460.6 + 32) =

*p*

_{0}V

_{0}τ/τ

_{0},

where τ, τ_{0} are the temperatures *t* and 32° reckoned from the absolute
zero, which is −460.6° Fahr.;

*p*/G =

*p*

_{0}τ/G

_{0}τ

_{0};

*a*)

_{0}G

_{0}/

*p*

_{0}τ.

If *p*_{0} = 2116.8, G_{0} = .08075, τ_{0} = 460.6 + 32 = 492.6, then

*p*/G = 53.2τ.

*a*)

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.

§ 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