§ 5. _Units of Volume._--In practical calculations the cubic foot and
gallon are largely used, and in metric countries the litre and cubic
metre (= 1000 litres). The imperial gallon is now exclusively used in
England, but the United States have retained the old English wine
gallon.
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 lb. per cub. ft., or 10 lb. 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 lb. 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 lb. 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 lb. 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 lb. 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,
/ / dV \ dp
k = dp / ( - -- ) = - V --.
/ \ V / dV
_Elasticity of Volume of Liquids._
+-----------+------------+-----------+------------+------------+
| | Canton. | Oersted. | Colladon | Regnault. |
| | | | and Sturm. | |
+-----------+------------+-----------+------------+------------+
| 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 [rho] 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._
+-------------+----------+----------+
| | | G |
| Temperature.| [rho] |Weight of |
+-----+-------+Density of|1 cub. ft.|
|Cent.| Fahr. | Water. | in lb. |
+-----+-------+----------+----------+
| 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
[rho], on the assumption that 1 cub. ft. of water at 39.2° Fahr. is
62.425 lb. 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 lb. per cub. ft., which is its density at 53° Fahr. It
may be noted also that ice at 32° Fahr. contains 57.3 lb. per cub. ft.
The values of [rho] 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 [omega] be the
area of section of the pipe, h the height of the pressure column, p
the intensity of pressure at P; then
p[omega] = Gh[omega] lb.,
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.