Part 19

(3) The values of these expressions for saturated vapors and for gases, at the same temperature, must be the same.

31. No conclusion can be drawn a priori regarding the values of this coefficient μ for different temperatures, which can only be determined, or compared, by experiment. The results of a great variety of experiments, in different branches of physical science (Pneumatics and Acoustics), cited by Carnot and by Clapeyron, indicate that the values of μ for low temperatures exceed the values for higher temperatures; a result amply verified by the continuous series of experiments performed by Regnault on the saturated vapor of water for all temperatures from 0° to 230°, which, as we shall see later, give values for μ gradually diminishing from the inferior limit to the superior limit of temperature. When, by observation, μ has been determined as a function of the temperature, the amount of mechanical effect, M, deducible from H units of heat descending from a body at the temperature S to a body at the temperature T, may be calculated from the expression

M = H ∫{S}^{T} μdt_, (7)

which is, in fact, what either of the equations (1) for the steam-engine, or (4) for the air-engine, becomes, when the notation μ, for Carnot’s multiplier, is introduced.

The values of this integral may be practically obtained, in the most convenient manner, by first determining, from observation, the mean values of μ for the successive degrees of the thermometric scale, and then adding the values for all the degrees within the limits of the extreme temperatures S and T.

32. The complete theoretical investigation of the motive power of heat is thus reduced to the experimental determination of the coefficient μ; and may be considered as perfect, when, by any series of experimental researches whatever, we can find a value of μ for every temperature within practical limits. The special character of the experimental researches, whether with reference to gases or with reference to vapors, necessary and sufficient for this object, is defined and restricted in the most precise manner, by the expressions (6) for μ, given above.

33. The object of Regnault’s great work, referred to in the title of this paper, is the experimental determination of the various physical elements of the steam-engine; and when it is complete, it will furnish all the data necessary for the calculation of μ. The valuable researches already published in a first part of that work make known the latent heat of a given weight, and the pressure, of saturated steam for all temperatures between 0° and 230° Cent. of the air-thermometer. Besides these data, however, the density of saturated vapor must be known, in order that k, the latent heat of a unit of volume, may be calculated from Regnault’s determination of the latent heat of a given weight. Between the limits of 0° and 100°, it is probable, from various experiments which have been made, that the density of vapor follows very closely the simple laws which are so accurately verified by the ordinary gases; and thus it may be calculated from Regnault’s table giving the pressure at any temperature within those limits. Nothing as yet is known with accuracy as to the density of saturated steam between 100 and 230°, and we must be contented at present to estimate it by calculation from Regnault’s table of pressures; although, when accurate experimental researches on the subject shall have been made, considerable deviations from the laws of Boyle and Dalton, on which this calculation is founded, may be discovered.

34. Such are the experimental data on which the mean values of μ for the successive degrees of the air-thermometer, from 0 to 230°, at present laid before the Royal Society, is founded. The unit of length adopted is the English foot; the unit of weight, the pound; the unit of work, a “foot-pound;” and the unit of heat that quantity which, when added to a pound of water at 0°, will produce an elevation of 1° in temperature. The mean value of μ for any degree is found to a sufficient degree of approximation by taking, in place of σ, dp/dt and k; in the expression

(1 − σ). (dp)/(kdt);

the mean values of those elements; or, what is equivalent to the corresponding accuracy of approximation, by taking, in place of σ and k respectively, the mean of the values of those elements for the limits of temperature, and in place of dp/dt, the difference of the values of p, at the same limits.

35. In Regnault’s work (at the end of the eighth memoir), a table of the pressures of saturated steam for the successive temperatures 0°, 1°, 2°, ... 230°, expressed in millimetres of mercury, is given. On account of the units adopted in this paper, these pressures must be estimated in pounds on the square foot, which we may do by multiplying each number of millimetres by 2.7896, the weight in pounds of a sheet of mercury, one millimetre thick, and a square foot in area.

36. The value of k, the latent heat of a cubic foot, for any temperature t, is found from λ, the latent heat of a pound of saturated steam, by the equation

k = (p)/(760). (1 + .00366 × 100)/(1 + .00366 × t). × .036869 . λ,

where p denotes the pressure in millimetres, and λ the latent heat of a pound of saturated steam; the values of λ being calculated by the empirical formula

λ = (606.5 + 0.305t) − (t + .00002t^2 + 0.0000003t^3),

given by Regnault as representing, between the extreme limits of his observations, the latent heat of a unit weight of saturated steam.

EXPLANATION OF TABLE I.

37. The mean values of μ for the first, for the eleventh, for the twenty-first, and so on, up to the 231st degree of the air-thermometer, have been calculated in the manner explained in the preceding paragraphs. These, and interpolated results, which must agree with what would have been obtained, by direct calculation from Regnault’s data, to three significant places of figures (and even for the temperatures between 0° and 100°, the experimental data do not justify us in relying on any of the results to a greater degree of accuracy), are exhibited in Table I.

To find the amount of mechanical effect due to a unit of heat, descending from a body at a temperature S to a body at T, if these numbers be integers, we have merely to add the values of μ in Table I. corresponding to the successive numbers.

T + 1, T + 2, ... S − 2, S − 1.

EXPLANATION OF TABLE II.

38. The calculation of the mechanical effect, in any case, which might always be effected in the manner described in § 37 (with the proper modification for fractions of degrees, when necessary), is much simplified by the use of Table II., where the first number of Table I., the sum of the first and second, the sum of the first three, the sum of the first four, and so on, are successively exhibited. The sums thus tabulated are the values of the integrals

∫{0}^1 μdt, ∫{0}^2 μdt, ∫{0}^3 μdt, ... ∫{0}^{231} μdt;

and, if we denote ∫{0}^t μdt by the letter M, Table II. may be regarded as a table of the value of M_.

To find the amount of mechanical effect due to a unit of heat descending from a body at a temperature S to a body at T, if these numbers be integers, we have merely to subtract the value of M, for the number T, from the value for the number S, given in Table II.

TABLE I. MEAN VALUES OF Μ FOR THE SUCCESSIVE DEGREES OF THE AIR-THERMOMETER FROM 0° TO 230°. ───────────────────────────────────┬─────────────────────────────────── ° │ μ ───────────────────────────────────┼─────────────────────────────────── 1│ 4.960 2│ 4.946 3│ 4.932 4│ 4.918 5│ 4.905 6│ 4.892 7│ 4.878 8│ 4.865 9│ 4.852 10│ 4.839 11│ 4.826 12│ 4.812 13│ 4.799 14│ 4.786 15│ 4.773 16│ 4.760 17│ 4.747 18│ 4.735 19│ 4.722 20│ 4.709 21│ 4.697 22│ 4.684 23│ 4.672 24│ 4.659 25│ 4.646 26│ 4.634 27│ 4.621 28│ 4.609 29│ 4.596 30│ 4.584 31│ 4.572 32│ 4.559 33│ 4.547 34│ 4.535 35│ 4.522 36│ 4.510 37│ 4.498 38│ 4.486 39│ 4.474 40│ 4.462 41│ 4.450 42│ 4.438 43│ 4.426 44│ 4.414 45│ 4.402 46│ 4.390 47│ 4.378 48│ 4.366 49│ 4.355 50│ 4.343 51│ 4.331 52│ 4.319 53│ 4.308 54│ 4.296 55│ 4.285 56│ 4.273 57│ 4.262 58│ 4.250 59│ 4.239 60│ 4.227 61│ 4.216 62│ 4.205 63│ 4.194 64│ 4.183 65│ 4.172 66│ 4.161 67│ 4.150 68│ 4.140 69│ 4.129 70│ 4.119 71│ 4.109 72│ 4.098 73│ 4.088 74│ 4.078 75│ 4.067 76│ 4.057 77│ 4.047 78│ 4.037 79│ 4.028 80│ 4.018 81│ 4.009 82│ 3.999 83│ 3.990 84│ 3.980 85│ 3.971 86│ 3.961 87│ 3.952 88│ 3.943 89│ 3.934 90│ 3.925 91│ 3.916 92│ 3.907 93│ 3.898 94│ 3.889 95│ 3.880 96│ 3.871 97│ 3.863 98│ 3.854 99│ 3.845 100│ 3.837 101│ 3.829 102│ 3.820 103│ 3.812 104│ 3.804 105│ 3.796 106│ 3.788 107│ 3.780 108│ 3.772 109│ 3.764 110│ 3.757 111│ 3.749 112│ 3.741 113│ 3.734 114│ 3.726 115│ 3.719 116│ 3.712 117│ 3.704 118│ 3.697 119│ 3.689 120│ 3.682 121│ 3.675 122│ 3.668 123│ 3.661 124│ 3.654 125│ 3.647 126│ 3.640 127│ 3.633 128│ 3.627 129│ 3.620 130│ 3.614 131│ 3.607 132│ 3.601 133│ 3.594 134│ 3.586 135│ 3.579 136│ 3.573 137│ 3.567 138│ 3.561 139│ 3.555 140│ 3.549 141│ 3.543 142│ 3.537 143│ 3.531 144│ 3.525 145│ 3.519 146│ 3.513 147│ 3.507 148│ 3.501 149│ 3.495 150│ 3.490 151│ 3.484 152│ 3.479 153│ 3.473 154│ 3.468 155│ 3.462 156│ 3.457 157│ 3.451 158│ 3.446 159│ 3.440 160│ 3.435 161│ 3.430 162│ 3.424 163│ 3.419 164│ 3.414 165│ 3.409 166│ 3.404 167│ 3.399 168│ 3.394 169│ 3.389 170│ 3.384 171│ 3.380 172│ 3.375 173│ 3.370 174│ 3.365 175│ 3.361 176│ 3.356 177│ 3.351 178│ 3.346 179│ 3.342 180│ 3.337 181│ 3.332 182│ 3.328 183│ 3.323 184│ 3.318 185│ 3.314 186│ 3.309 187│ 3.304 188│ 3.300 189│ 3.295 190│ 3.291 191│ 3.287 192│ 3.282 193│ 3.278 194│ 3.274 195│ 3.269 196│ 3.265 197│ 3.261 198│ 3.257 199│ 3.253 200│ 3.249 201│ 3.245 202│ 3.241 203│ 3.237 204│ 3.233 205│ 3.229 206│ 3.225 207│ 3.221 208│ 3.217 209│ 3.213 210│ 3.210 211│ 3.206 212│ 3.202 213│ 3.198 214│ 3.195 215│ 3.191 216│ 3.188 217│ 3.184 218│ 3.180 219│ 3.177 220│ 3.173 221│ 3.169 222│ 3.165 223│ 3.162 224│ 3.158 225│ 3.155 226│ 3.151 227│ 3.148 228│ 3.144 229│ 3.141 230│ 3.137 231│ 3.134 ───────────────────────────────────┴───────────────────────────────────

TABLE II. MECHANICAL EFFECT IN FOOT-POUNDS DUE TO A THERMIC UNIT CENTIGRADE, PASSING FROM A BODY, AT ANY TEMPERATURE LESS THAN 230° TO A BODY AT 0°. ───────────────────────────────────┬─────────────────────────────────── Superior Limit of Temperature. │ Mechanical Effect. ───────────────────────────────────┼─────────────────────────────────── ° │ Ft.-Pounds. │ 1│ 4.960 2│ 9.906 3│ 14.838 4│ 19.756 5│ 24.661 6│ 29.553 7│ 34.431 8│ 39.296 9│ 44.148 10│ 48.987 11│ 53.813 12│ 58.625 13│ 63.424 14│ 68.210 15│ 72.983 16│ 77.743 17│ 82.490 18│ 87.225 19│ 91.947 20│ 96.656 21│ 101.353 22│ 106.037 23│ 110.709 24│ 115.368 25│ 120.014 26│ 124.648 27│ 129.269 28│ 133.878 29│ 138.474 30│ 143.058 31│ 147.630 32│ 152.189 33│ 156.736 34│ 161.271 35│ 165.793 36│ 170.303 37│ 174.801 38│ 179.287 39│ 183.761 40│ 188.223 41│ 192.673 42│ 197.111 43│ 201.537 44│ 205.951 45│ 210.353 46│ 214.743 47│ 219.121 48│ 223.487 49│ 227.842 50│ 232.185 51│ 236.516 52│ 240.835 53│ 245.143 54│ 249.439 55│ 253.724 56│ 257.997 57│ 262.259 58│ 266.509 59│ 270.748 60│ 274.975 61│ 279.191 62│ 283.396 63│ 287.590 64│ 291.773 65│ 295.945 66│ 300.106 67│ 304.256 68│ 308.396 69│ 312.525 70│ 316.644 71│ 320.752 72│ 324.851 73│ 328.939 74│ 333.017 75│ 337.084 76│ 341.141 77│ 345.188 78│ 349.225 79│ 353.253 80│ 357.271 81│ 361.280 82│ 365.279 83│ 369.269 84│ 373.249 85│ 377.220 86│ 381.181 87│ 385.133 88│ 389.076 89│ 393.010 90│ 396.935 91│ 400.851 92│ 404.758 93│ 408.656 94│ 412.545 95│ 416.425 96│ 420.296 97│ 424.159 98│ 428.013 99│ 431.858 100│ 435.695 101│ 439.524 102│ 443.344 103│ 447.156 104│ 450.960 105│ 454.756 106│ 458.544 107│ 462.324 108│ 466.096 109│ 469.860 110│ 473.617 111│ 477.366 112│ 481.107 113│ 484.841 114│ 488.567 115│ 492.286 116│ 495.998 117│ 499.702 118│ 503.399 119│ 507.088 120│ 510.770 121│ 514.445 122│ 518.113 123│ 521.174 124│ 525.428 125│ 529.075 126│ 532.715 127│ 536.348 128│ 539.975 129│ 543.595 130│ 547.209 131│ 550.816 132│ 554.417 133│ 558.051 134│ 561.597 135│ 565.176 136│ 568.749 137│ 572.316 138│ 575.877 139│ 579.432 140│ 582.981 141│ 586.524 142│ 590.061 143│ 593.592 144│ 597.117 145│ 600.636 146│ 604.099 147│ 607.656 148│ 611.157 149│ 614.652 150│ 618.142 151│ 621.626 152│ 625.105 153│ 628.578 154│ 632.046 155│ 635.508 156│ 638.965 157│ 642.416 158│ 645.862 159│ 649.302 160│ 652.737 161│ 656.167 162│ 659.591 163│ 663.010 164│ 666.424 165│ 669.833 166│ 673.237 167│ 676.636 168│ 680.030 169│ 683.419 170│ 686.803 171│ 690.183 172│ 693.558 173│ 696.928 174│ 700.293 175│ 703.654 176│ 707.010 177│ 710.361 178│ 713.707 179│ 717.049 180│ 720.386 181│ 723.718 182│ 727.046 183│ 730.369 184│ 733.687 185│ 737.001 186│ 740.310 187│ 743.614 188│ 746.914 189│ 750.209 190│ 753.500 191│ 756.787 192│ 760.069 193│ 763.347 194│ 766.621 195│ 769.890 196│ 773.155 197│ 776.416 198│ 779.673 199│ 782.926 200│ 786.175 201│ 789.420 202│ 792.661 203│ 795.898 204│ 799.131 205│ 802.360 206│ 805.585 207│ 808.806 208│ 812.023 209│ 815.236 210│ 818.446 211│ 821.652 212│ 824.854 213│ 828.052 214│ 831.247 215│ 834.438 216│ 837.626 217│ 840.810 218│ 843.990 219│ 847.167 220│ 850.340 221│ 853.509 222│ 856.674 223│ 859.836 224│ 862.994 225│ 866.149 226│ 869.300 227│ 872.448 228│ 875.592 229│ 878.733 230│ 881.870 231│ 885.004 ───────────────────────────────────┴───────────────────────────────────

Note on the curves described in Clapeyron’s graphical method of exhibiting Carnot’s Theory of the Steam-Engine.