460 HYDROMECHANICS [HYDRAULICS. on the general velocities of translation at different points of the fluid (or what M. Boussinesq terms the mean local velo cities), but rather on the intensity at each point of the eddy ing agitation. The problems of hydraulics are therefore much more complicated than problems in which a regular motion of the fluid is assumed, hindered by the viscosity of the fluid. RELATION OF PRESSURE, DENSITY, AND TEMPERATURE OF LIQUIDS. 5. Density of Water. Water at ordinary temperature and pres sure contains 62 4 It per cubic foot, or 1000 kilogrammes per cubic metre. The density or weight per unit of volume will be designated by G. Elver and spring water is not sensibly denser than pure water, being at most l-100000th heavier. Sea-water may be taken at 64 ft per cubic foot. 6. Compressibility of Liquids. The most accurate experiments show that liquids are sensibly compressed by very great pressures, and tjiat up to a pressure of 65 atmospheres, or about 1000 Ib per square inch, the compression is proportional to the pressure. The chief results of experiment are given in the following table. Let V : be the volume of a liquid in cubic feet under a pressure p l Ib pel- square foot, and V 2 its volume under a pressure p y Then the cubical compression is and the ratio of the increase of pressure p. - p l to the cubical com- I V} ~- v T pression is sensibly constant. That is, r?_ri^n. j s constant.
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This constant is termed the elasticity of volume, and is denoted by k (Thomson). With the notation of the differential calculus, dV V Elasticity of Volume of Liquids. Canton. Ocrstedt. Collation and Sturm. Kegnault. Water Sea water . . . Mercury Oil 45,990,000 52,900,000 705,300,000 44,090,000 45,900,000 42,660,000 626,100,000 44,090,000 604,500,000 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 Tem perature. Although the change of volume of water with change of temperature is so small that it may generally be neglected in ordi nary hydraulic calculations, yet it should be noted that there is a change of volume which should be allowed for in very exact calcu lations. The values of p 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. p Density of Water. G Weight of 1 c. ft. in Ib. Temperature. p Density of Water. G Weight of 1 c. ft. in Ib. Cent. Fahr. Cent. Fahr.
32-0 999884 62-417 20 68-0 998272 62-316 1 33-8 999941 62-420 22 71-6 997839 62 "289 2 35-6 999982 62-423 24 7-2 997380 62-261 3 37-4 1-000004 62-424 26 78-8 996879 62-229 4 39-2 1-000013 62-425 28 82-4 996344 62-196 5 41-0 1 -000003 62-424 30 86 995778 62-161 6 42-8 999983 62-423 35 95 99469 62-093 7 44 6 999946 62-421 40 104 99236 61-947 8 46-4 999899 62-418 45 113 99038 61 -823 9 48-2 999837 62-414 50 122 98821 61-688 10 50-0 999760 62-409 55 131 98583 61 -540 11 51-8 999668 62-403 60 140 98339 61-387 12 53-6 999562 62-397 65 149 98075 61-222 13 55-4 999443 62-389 70 158 97795 61-048 14 57-2 999312 62-381 75 167 97499 60-863 15 59-0 999173 62-373 80 176 97195 60-674 16 60-8 999015 62-363 85 185 96880 60-477 17 62-6 998854 62-353 90 194 96557 60-275 18 64-4 998667 62-341 100 212 95866 59-844 19 66-2 998473 62-329 The weight per cubic foot lias been calculated from the values of p, on the assumption that a cubic foot of water at 39 2 3 Fahr. is 62 425 Ib. 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 Ib per cubic foot, which is its density at 53 Fahr. It may be noted also that ice at 32 Fahr. contains 57 - 2 Ib per cubic foot. The values of p^ are the densities in grammes per cubic centimetre. 8. Pressure Column. Free Surface Level. Suppose a small verti cal pipe introduced into a liquid at any point P (tig. 13). Then the liquid will rise in the pipe to a level 0, such that the pres- suredue 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 ac curately parallel to the direction of mo tion, or the impact of the liquid at the mouth of the pipe will have an influence Fig. 13. 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 u be the area of section of the pipe, h the height of the pressure column, p the intensity of pressure at P ; then pta = Gha) Ib ,
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that is, h is the height due to the pressure at p. The level OC will be termed the free surface level corresponding to the pressure at P. RELATION OF PRESSURE, TEMPERATURE, AND DENSITY OF GASES. 9. Relation of Pressure, Volume, Temperature, and Density in Com pressible Fluids. Certain problems on the flow of air and steam are so similar to those relating to the flow of water that they are con veniently 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). Letjt> be mean atmospheric pressure (21 16 8 Ib per square foot), V c the volume of 1 Ib of air at 32 Fahr. under the pressure p . Then ....... (1). If GU is the weiglit per cubic foot of air in the same conditions, 26214 (2). For any other pressure p, at which the volume of 1 tt> is V and the weight per cubic foot is G, the temperature being 32 Fahr. , V S6214 OTG =2iln (3). Change of Pressure or Volume by Change of Temperature. Let p , V , G , as before be the pressure, the volume of a pound in cubic feet, and the weiglit of a cubic foot in pounds, at 32 Fahr. Let^>, 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, 460-6-M . . (4), where T, T O are the temperatures t and 32 reckoned from the abso lute zero, which is -460 6 Fahr. ; Po G; Po = 2116-8, G = 08075, T O = 460 6 + 32 = 492 6, then (5).
(5o).
