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 μ_dt_, ∫_{0}^2 μ_dt_, ∫_{0}^3 μ_dt_, ... ∫_{0}^{231} μ_dt_;
and, if we denote ∫_{0}^t μ_dt_ 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.[56] 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._
39. At any instant when the temperature of the water and vapor is _t_, during the fourth operation (see above, § 16, and suppose, for the sake of simplicity, that at the beginning of the first and at the end of the fourth operation the piston is absolutely in contact with the surface of the water), the latent heat of the vapor must be precisely equal to the amount of heat that would be necessary to raise the temperature of the whole mass, if in the liquid state, from _t_ to _S_.[57] Hence, if _v′_ denote the volume of the vapor, _c_ the mean capacity for heat of a pound of water between the temperatures _S_ and _t_, and _W_ the weight of the entire mass, in pounds, we have
_kv′_ = _c_(_S_ − _t_)_W_.
Again, the circumstances during the second operation are such that the mass of liquid and vapor possesses _H_ units of heat more than during the fourth; and consequently, at the instant of the second operation, when the temperature is _t_, the volume _v_ of the vapor will exceed _v′_ by an amount of which the latent heat is _H_, so that we have
_v_ = _v′_ + (_H_)/(_k_).
40. Now, at any instant, the volume between the piston and its primitive position is less than the actual volume of vapor by the volume of the water evaporated. Hence, if _x_ and _x′_ denote the abscissæ of the curve at the instants of the second and fourth operations respectively, when the temperature is _t_, we have
_x_ = _v_ − σ_v_, _x′_ = _v′_ − σ_v′_,
and, therefore, by the preceding equations,
_x_ = (1 − σ)/(_k_){_H_ + _c_(_S_ − _t_)_W_}, (_a_) _x′_ = (1 − σ)/(_k_)_c_(_S_ − _t_)_W_. (_b_) These equations, along with _y_ = _y′_ = _p_, (_c_)
enable us to calculate, from the data supplied by Regnault, the abscissa and ordinate for each of the curves described above (§ 17) corresponding to any assumed temperature _t_. After the explanations of §§ 33, 34, 35, 36, it is only necessary to add that _c_ is a quantity of which the value is very nearly unity, and would be exactly so were the capacity of water for heat the same at every temperature as it is between 0° and 1°; and that the value of _c_(_S_ − _t_), for any assigned values of _S_ and _t_, is found, by subtracting the number corresponding to _t_ from the number corresponding to _s_, in the column headed “_Nombre des unités de chaleur abandonnées par un kilogramme d’eau en descendant de T° à 0°_,” of the last table (at the end of the tenth memoir) of Regnault’s work. By giving _S_ the value 230°, and by substituting successively 220, 210, 200, etc., for _t_, values for _x_, _y_, _x′_, _y′_, have been found, which are exhibited in the table opposite.
─────────────┬─────────────────┬────────────────────────┬───────────── Temperatures.│ Volumes to be │ Volumes from the │Pressures of │described by the │ primitive position of │ saturated │ piston, to │ the piston to those │ steam, in │ complete the │occupied at instants of │pounds on the │fourth operation.│ the second operation. │square foot. _t_ │ _x′_ │ _x_ │_y_ = _y′_ = │ │ │ _p_ ─────────────┼─────────────────┼────────────────────────┼───────────── 0°│ 1269. _W_ │_x′_ + 5.409._H_ │ 12.832 10│ 639.6. _W_ │_x′_ + 2.847._H_ │ 25.567 20│ 337.3. _W_ │_x′_ + 1.571._H_ │ 48.514 30│ 185.5. _W_ │_x′_ + .9062._H_ │ 88.007 40│ 105.9. _W_ │_x′_ + .5442._H_ │ 153.167 50│ 62.62. _W_ │_x′_ + .3392._H_ │ 256.595 60│ 38.19. _W_ │_x′_ + .2188._H_ │ 415.070 70│ 21.94. _W_ │_x′_ + .1456._H_ │ 650.240 80│ 15.38. _W_ │_x′_ + .09962._H_ │ 989.318 90│ 10.09. _W_ │_x′_ + .06994._H_ │ 1465.80 100│ 6.744. _W_ │_x′_ + .05026._H_ │ 2120.11 110│ 4.578. _W_ │_x′_ + .03688._H_ │ 2999.87 120│ 3.141. _W_ │_x′_ + .02758._H_ │ 4160.10 130│ 2.176. _W_ │_x′_ + .02098._H_ │ 5663.70 140│ 1.519. _W_ │_x′_ + .01625._H_ │ 7581.15 150│ 1.058. _W_ │_x′_ + .01271._H_ │ 9990.26 160│ 0.7369. _W_ │_x′_ + .01010._H_ │ 12976.2 170│ 0.5085. _W_ │_x′_ + .008116._H_ │ 16630.7 180│ 0.3454. _W_ │_x′_ + .006592._H_ │ 21051.5 190│ 0.2267. _W_ │_x′_ + .005406._H_ │ 26341.5 200│ 0.1409. _W_ │_x′_ + .004472._H_ │ 32607.7 210│ 0.0784. _W_ │_x′_ + .003729._H_ │ 39960.7 220│ 0.3310. _W_ │_x′_ + .003130._H_ │ 48512.4 230│ 0 │_x′_ + .002643._H_ │ 58376.6 ─────────────┴─────────────────┴────────────────────────┴─────────────
_Appendix._
(Read April 30, 1849.)
41. In p. 30 some conclusions drawn by Carnot from his general reasoning were noticed; according to which it appears, that if the value of μ for any temperature is known, certain information may be derived with reference to the saturated vapor of any liquid whatever, and, with reference to any gaseous mass, without the necessity of experimenting upon the specific medium considered. Nothing in the whole range of Natural Philosophy is more remarkable than the establishment of general laws by such a process of reasoning. We have seen, however, that doubt may exist with reference to the truth of the axiom on which the entire theory is founded, and it therefore becomes more than a matter of mere curiosity to put the inferences deduced from it to the test of experience. The importance of doing so was clearly appreciated by Carnot; and, with such data as he had from the researches of various experimenters, he tried his conclusions. Some very remarkable propositions which he derives from his theory coincide with Dulong and Petit’s subsequently discovered experimental laws with reference to the heat developed by the compression of a gas; and the experimental verification is therefore in this case (so far as its accuracy could be depended upon) decisive. In other respects, the data from experiment were insufficient, although, so far as they were available as tests, they were confirmatory of the theory.
42. The recent researches of Regnault add immensely to the experimental data available for this object, by giving us the means of determining with considerable accuracy the values of μ within a very wide range of temperature, and so affording a trustworthy standard for the comparison of isolated results at different temperatures, derived from observations in various branches of physical science.
In the first section of this Appendix the theory is tested, and shown to be confirmed by the comparison of the values of μ found above, with those obtained by Carnot and Clapeyron from the observations of various experimenters on air, and the vapors of different liquids. In the second and third sections some striking confirmations of the theory arising from observations by Dulong, on the specific heat of gases, and from Mr. Joule’s experiments on the heat developed by the compression of air, are pointed out; and in conclusion, the actual methods of obtaining mechanical effect from heat are briefly examined with reference to their economy.
I. _On the values of μ derived by Carnot and Clapeyron from observations on Air, and on the Vapors of various liquids._
43. In Carnot’s work, pp. 80–82, the mean value of μ between 0° and 1° is derived from the experiments of Delaroche and Bérard on the specific heat of gases, by a process approximately equivalent to the calculation of the value of (_Ep_{0}v_{0}_)/(_vdq_/_dv_) for the temperature ½°. There are also, in the same work, determinations of the values of μ from observations on the vapors of alcohol and water; but a table given in M. Clapeyron’s paper, of the values of μ derived from the data supplied by various experiments with reference to the vapors of ether, alcohol, water, and oil of turpentine, at the respective boiling-points of these liquids, affords us the means of comparison through a more extensive range of temperature. In the cases of alcohol and water, these results ought of course to agree with those of Carnot. There are, however, slight discrepancies which must be owing to the uncertainty of the experimental data.[58] In the opposite table, Carnot’s results with reference to air, and Clapeyron’s results with reference to the four different liquids, are exhibited, and compared with the values of μ which have been given above (Table I.) for the same temperatures, as derived from Regnault’s observations on the vapor of water.
────────────┬──────────────┬──────────────┬──────────────┬───────────── │ │ │ Values of μ │ │ │ │ deduced from │ Names of the│ │ │ Regnault’s │ Media. │Temperatures. │ Values of μ. │Observations. │Differences. ────────────┼──────────────┼──────────────┼──────────────┼───────────── │ ° │ (Carnot) │ │ Air │ 0.5│ 4.377│ 4.960│ .383 Sulphuric │ (Boil. pt.)│ (Clapeyron)│ │ Ether │ 35.5│ 4.478│ 4.510│ .032 Alcohol │ 78.8│ 3.963│ 4.030│ .071 Water │ 100│ 3.658│ 3.837│ .179 Essence of │ │ │ │ Turpentine│ 156.8│ 3.530│ 3.449│ −.081 ────────────┴──────────────┴──────────────┴──────────────┴─────────────
44. It may be observed that the discrepancies between the results founded on the experimental data supplied by the different observers with reference to water at the boiling-point, are greater than those which are presented between the results deduced from any of the other liquids, and water at the other temperatures; and we may therefore feel perfectly confident that the verification is complete to the extent of accuracy of the observations.[59] The considerable discrepancy presented by Carnot’s result deduced from experiments on air, is not to be wondered at when we consider the very uncertain nature of his data.
45. The fact of the gradual decrease of μ through a very extensive range of temperature, being indicated both by Regnault’s continuous series of experiments and by the very varied experiment on different media, and in different branches of Physical Science, must be considered as a striking verification of the theory.
II. _On the Heat developed by the Compression of Air._
46. Let a mass of air, occupying initially a given volume _V_, under a pressure _P_, at a temperature _t_, be compressed to a less volume _V′_, and allowed to part with heat until it sinks to its primitive temperature _t_. The quantity of heat which is evolved may be determined, according to Carnot’s theory, when the particular value of μ, corresponding to the temperature _t_, is known. For, by § 30, equation (6), we have
_v_(_dq_)/(_dv_) = (_Ep_{0}v_{0}_)/(μ),
where _dq_ is the quantity of heat absorbed, when the volume is allowed to increase from _v_ to _v_ + _dv_; or the quantity evolved by the reverse operation. Hence we deduce
_dq_ = (_Ep_{0}v_{0}_)/(μ) (_dv_)/(_v_). (8)
Now, (_Ep_{0}v_{0}_)/(μ) is constant, since the temperature remains unchanged; and therefore we may at once integrate the second number. By taking it between the limits _V′_ and _V_, we thus find
_Q_ = (_Ep_{0}v_{0}_)/(μ) log (_V_)/(_V′_)[60], (9)
where _Q_ denotes the required amount of heat evolved by the compression from _V_ to _P′_. This expression may be modified by employing the equations _PV_ = _P′V′_ = _p_{0}v_{0}_(1 + _Et_); and we thus obtain
_Q_ = (_EPV_)/(μ(1 + _Et_)) log (_V_)/(_V′_) = (_EP′V′_)/(μ(1 + _Et_)) log (_V_)/(_V′_). (10)
From this result we draw the following conclusion:
47. _Equal volumes of all elastic fluids, taken at the same temperature and pressure, when compressed to smaller equal volumes, disengage equal quantities of heat._
This extremely remarkable theorem of Carnot’s was independently laid down as a probable experimental law by Dulong, in his “_Recherches sur la Chaleur Spécifique des Fluides Élastiques_,” and it therefore affords a most powerful confirmation of the theory.[61]
48. In some very remarkable researches made by Mr. Joule upon the heat developed by the compression of air, the quantity of heat produced in different experiments has been ascertained with reference to the amount of work spent in the operation. To compare the results which he has obtained with the indications of theory, let us determine the amount of work necessary actually to produce the compression considered above.
49. In the first place, to compress the gas from the volume _v_ + _dv_ to _v_, the work required is _pdv_, or, since
_pv_ = _p_{0}v_{0}_(1 + _Et_), _p_{0}v_{0}_(1 + _Et_)(_dv_)/(_v_).
Hence, if we denote by _W_ the total amount of work necessary to produce the compression from _V_ to _V′_, we obtain, by integration,
_W_ = _p_{0}v_{0}_(1 + _Et_) log (_V_)/(_V′_).
Comparing this with the expression above, we find
(_W_)/(_Q_) = (μ(1 + _Et_))/(_E_). (11)
50. Hence we infer that—
(1) The amount of work necessary to produce a unit of heat by the compression of a gas is the same for all gases at the same temperature;
(2) And that the quantity of heat evolved in all circumstances, when the temperature of the gas is given, is proportional to the amount of work spent in the compression.
51. The expression for the amount of work necessary to produce a unit of heat is
μ(1 + _Et_)/(_E_),
and therefore Regnault’s experiments on steam are available to enable us to calculate its value for any temperature. By finding the values of μ at 0°, 10°, 20°, etc., from Table I., and by substituting successively the values 0, 10, 20, etc., for _t_, the following results have been obtained:
TABLE OF THE VALUES OF (μ(1 + _Et_))/(_E_). ───────────────────────────────────┬─────────────────────────────────── Work requisite to produce a unit of│ Temperature of the Gas. Heat by the compression of a Gas. │ ───────────────────────────────────┼─────────────────────────────────── Ft.-pounds. │ ° 1357.1 │ 0 1368.7 │ 10 1379.0 │ 20 1388.0 │ 30 1395.7 │ 40 1401.8 │ 50 1406.7 │ 60 1412.0 │ 70 1417.6 │ 80 1424.0 │ 90 1430.6 │ 100 1438.2 │ 110 1446.4 │ 120 1455.8 │ 130 1465.3 │ 140 1475.8 │ 150 1489.2 │ 160 1499.0 │ 170 1511.3 │ 180 1523.5 │ 190 1536.5 │ 200 1550.2 │ 210 1564.0 │ 220 1577.8 │ 230 ───────────────────────────────────┴───────────────────────────────────
Mr. Joule’s experiments were all conducted at temperatures from 50° to about 60° Fahr., or from 10° to 16° Cent.; and consequently, although some irregular differences in the results, attributable to errors of observation inseparable from experiments of such a very difficult nature, are presented, no regular dependence on the temperature is observable. From three separate series of experiments, Mr. Joule deduces the following numbers for the work, in foot-pounds, necessary to produce a thermic unit Fahrenheit by the compression of a gas.
820, 814, 760.
Multiplying these by 1.8, to get the corresponding number for a thermic unit Centigrade, we
1476, 1465, and 1368.
The largest of these numbers is most nearly conformable with Mr. Joule’s views of the relation between such experimental “equivalents,” and others which he obtained in his electro-magnetic researches; but the smallest agrees almost perfectly with the indications of Carnot’s theory; from which, as exhibited in the preceding table, we should expect, from the temperature in Mr. Joule’s experiments, to find a number between 1369 and 1379 as the result.[62]
III. _On the Specific Heats of Gases._
52. The following proposition is proved by Carnot as a deduction from his general theorem regarding the specific heats of gases.
_The excess of specific heat[63] under a constant pressure above the specific heat at a constant volume, is the same for all gases at the same temperature and pressure._
53. To prove this proposition, and to determine an expression for the “excess” mentioned in its enunciation, let us suppose a unit of volume of a gas to be elevated in temperature by a small amount, τ. The quantity of heat required to do this will be _A_τ, if _A_ denote the specific heat at a constant volume. Let us next allow the gas to expand without going down in temperature, until its pressure becomes reduced to its primitive value. The expansion which will take place will be (_E_τ)/(1 + _Et_), if the temperature be denoted by _t_; and hence, by (8), the quantity of heat that must be supplied, to prevent any lowering of temperature, will be
(_Ep_{0}v_{0}_)/(μ) . (_E_τ)/(1 + _Et_), or (_E^2p_)/(μ(1 + _Et_)^2)τ.
Hence the total quantity added is equal to
_Α_τ + (_E^2p_)/(μ(1 + _Et_)^2)τ.
But, since _B_ denotes the specific heat under constant pressure, the quantity of heat requisite to bring the gas into this state, from its primitive condition, is equal to _Β_τ, and hence we have
_B_ = _A_ + (_E^2p_)/(μ(1 + _Et_)^2). (12)
IV. _Comparison of the Relative Advantages of the Air-engine and Steam-engine._
54. In the use of water-wheels for motive power, the economy of the engine depends not only upon the excellence of its adaptation for actually transmitting any given quantity of water through it, and producing the equivalent of work, but upon turning to account the entire available fall; so, as we are taught by Carnot, the object of a thermodynamic engine is to economize in the best possible way the transference of all the heat evolved, from bodies at the temperature of the source, to bodies at the lowest temperature at which the heat can be discharged. With reference, then, to any engine of the kind, there will be two points to be considered:
(1) The extent of the _fall_ utilized.
(2) The economy of the engine, with the fall which it actually uses.
55. In the first respect, the air-engine, as Carnot himself points out, has a vast advantage over the steam-engine; since the temperature of the hot part of the machine may be made very much higher in the air-engine than would be possible in the steam-engine, on account of the very high pressure produced in the boiler, by elevating the temperature of the water which it contains to any considerable extent above the atmospheric boiling-point. On this account a “perfect air-engine” would be a much more valuable instrument than a “perfect steam-engine.”[64]
Neither steam-engines nor air-engines, however, are nearly perfect; and we do not know in which of the two kinds of machine the nearest approach to perfection may be actually attained. The beautiful engine invented by Mr. Stirling of Galston may be considered as an excellent beginning for the air-engine;[65] and it is only necessary to compare this with Newcomen’s steam-engine, and consider what Watt has effected, to give rise to the most sanguine anticipations of improvement.
V. _On the Economy of Actual Steam-engines._
56. The steam-engine being universally employed at present as the means for deriving motive power from heat, it is extremely interesting to examine, according to Carnot’s theory, the economy actually attained in its use. In the first place we remark, that out of the entire “fall” from the temperature of the coals to that of the atmosphere it is only part—that from the temperature of the boiler to the temperature of the condenser—that is made available; while the very great fall from the temperature of the burning coals to that of the boiler, and the comparatively small fall from the temperature of the condenser to that of the atmosphere, are entirely lost as far as regards the mechanical effect which it is desired to obtain. We infer from this, that the temperature of the boiler ought to be kept as high as, according to the strength, is consistent with safety, while that of the condenser ought to be kept as nearly down at the atmospheric temperature as possible. To take the entire benefit of the actual fall, Carnot showed that the “principle of expansion” must be pushed to the utmost.[66]
57. To obtain some notion of the economy which has actually been obtained, we may take the alleged performances of the best Cornish engines, and some other interesting practical cases, as examples.[67]
(1) The engine of _the Fowey Consols mine_ was reported, in 1845, to have given 125,089,000 foot-pounds of effect, for the consumption of one bushel or 94 lbs. of coals. Now the average amount evaporated from Cornish boilers, by one pound of coal, is 8½ lbs. of steam; and hence for each pound of steam evaporated 156,556 foot-pounds of work are produced.
The pressure of the saturated steam in the boiler may be taken as 3½ atmospheres;[68] and, consequently, the temperature of the water will be 140°. Now (Regnault, end of Mémoire X.) the latent heat of a pound of saturated steam at 140° is 508, and since, to compensate for each pound of steam removed from the boiler in the working of the engine, a pound of water, at the temperature of the condenser, which may be estimated at 30°, is introduced from the hot-well; it follows that 618 units of heat are introduced to the boiler for each pound of water evaporated. But the work produced, for each pound of water evaporated, was found above to be 156,556 foot-pounds. Hence ¹⁵⁶⁵⁵⁶⁄₆₁₈, or 253 foot-pounds, is the amount of work produced for each unit of heat transmitted through the Fowey Consols engine. Now in Table II. we find 583.0 as the theoretical effect due to a unit descending from 140° to 0°, and 143 as the effect due to a unit descending from 30° to 0°. The difference of these numbers, or 440,[69] is the number of foot-pounds of work that a _perfect_ engine with its boiler at 140° and its condenser at 30° would produce for each unit of heat transmitted. Hence the Fowey Consols engine, during the experiments reported on, performed ²⁵³⁄₄₄₀ of its theoretical duty, or 57½ per cent.
(2) The best duty on record, as performed by an engine at work (not for merely experimental purposes), is that of Taylor’s engine, at the United Mines, which in 1840 worked regularly for several months at the rate of 98,000,000 foot-pounds for each bushel of coals burned. This is ⁹⁸⁄₁₂₅, or .784 of the experimental duty reported in the case of the Fowey Consols engine. Hence the best useful work on record is at the rate of 198.3 foot-pounds for each unit of heat transmitted, and is (198.3)/(440) or 45 per cent of the theoretical duty, on the supposition that the boiler is at 140° and the condenser at 30°.
(3) French engineers contract (in Lille, in 1847, for example) to make engines for mill-power which will produce 30,000 metre-pounds or 98,427 foot-pounds of work for each pound of steam used. If we divide this by 618, we find 159 foot-pounds for the work produced by each unit of heat. This is 36.1 per cent of 440, the theoretical duty.[70]
(4) English engineers have contracted to make engines and boilers which will require only 3⅓ lbs. of the best coal per horse-power per hour. Hence in such engines each pound of coal ought to produce 565,700 foot-pounds of work, and if 7 lbs. of water be evaporated by each pound of coal, there would result 83,814 foot-pounds of work for each pound of water evaporated. If the pressure in the boiler be 3½ atmospheres (temperature 140°) the amount of work for each unit of heat will be found, by dividing this by 618, to be 130.7 foot-pounds, which is (130.7)/(440) or 29.7 per cent of the theoretical duty.[71]
(5) The actual average of work performed by good Cornish engines and boilers is 55,000,000 foot-pounds for each bushel of coal, or less than half the experimental performance of the Fowey Consols engine, more than half the actual duty performed by the United Mines engine in 1840; in fact, about 25 per cent of the theoretical duty.
(6) The average performances of a number of Lancashire engines and boilers have been recently found to be such as to require 12 lbs. of Lancashire coal per horse-power per hour (i.e., for performing 60 × 33,000 foot-pounds), and of a number of Glasgow engines such as to require 15 lbs. (of a somewhat inferior coal) for the same effect. There are, however, more than twenty large engines in Glasgow at present[72] which work with a consumption of only 6½ lbs. of dross, equivalent to 5 lbs. of the best Scotch or 4 lbs. of the best Welsh coal, per horse-power per hour. The economy may be estimated from these data, as in the other cases, on the assumption which, with reference to these, is the most probable we can make, that the evaporation produced by a pound of best coal is 7 lbs. of steam.
58. The following tables afford a synoptic view of the performances and theoretical duties in the various cases discussed above.
In Table A the numbers in the second column are found by dividing the numbers in the first by 8½ in cases (1), (2), and (5), and by 7 in cases (4), (6), and (7), the estimated numbers of pounds of steam actually produced in the different boilers by the burning of 1 lb. of coal.
The numbers in the third column are found from those in the second, by dividing by 618 in Table A, and 614 in Table B, which are respectively the quantities of heat required to convert a pound of water taken from the hot-well at 30°, into saturated steam, in the boiler, at 140° or at 121°.
With reference to the cases (3), (4), (6), (7), the hypothesis of Table B is probably in general nearer the truth than that of Table A. In (4), (6), and (7), especially upon hypothesis B, there is much uncertainty as to the amount of evaporation that will be actually produced by 1 lb. of fuel. The assumption on which the numbers in the second column in Table B are calculated, is, that each pound of coal will send the same number of units of heat into the boiler, whether hypothesis A or hypothesis B be followed. Hence, except in the case of the French contract, in which the _evaporation_, not the fuel, is specified, the numbers in the third column are the same as those in the third column of Table A.
TABLE A. VARIOUS ENGINES IN WHICH THE TEMPERATURE OF THE BOILER IS 140° C. AND THAT OF THE CONDENSER 30° C. _Theoretical Duty for each Unit of Heat transmitted, 440[73] foot-pounds._ ─────────────────┬─────────────┬──────────────┬─────────────┬────────── CASES. │Work produced│Work produced │Work produced│Percentage │for each lb. │ for each lb. │for each unit│ of │ of coal │ of water │ of heat │theoretical │ consumed. │ evaporated. │transmitted. │ duty. ─────────────────┼─────────────┼──────────────┼─────────────┼────────── │ Ft.-lbs. │ Ft.-lbs. │ Ft.-lbs. │ (1) Fowey Consols│ │ │ │ experiment,│ 1,330,734│ 156,556│ 253│ 57.5 reported in│ │ │ │ 1845 │ │ │ │ (2) Taylor’s │ │ │ │ engine at │ │ │ │ the United │ 1,042,553│ 122,653│ 198.4│ 45.1 Mines, │ │ │ │ working in │ │ │ │ 1840 │ │ │ │ (3) French │ │ │ │ engines, │ │ 98,427│ 159│ 36.1 according │ │ │ │ to contract│ │ │ │ (4) English │ │ │ │ engines, │ 565,700│ 80,814│ 130.8│ 29.7 according │ │ │ │ to contract│ │ │ │ (5) Average │ │ │ │ actual │ │ │ │ performance│ 585,106│ 68,836│ 111.3│ 25.3 of Cornish │ │ │ │ engines │ │ │ │ (6) Common │ │ │ │ engines, │ │ │ │ consuming │ │ │ │ 12 lbs. of │ 165,000│ 23,571│ 38.1│ 8.6 best coal │ │ │ │ per │ │ │ │ horse-power│ │ │ │ per hour │ │ │ │ (7) Improved │ │ │ │ engines │ │ │ │ with │ │ │ │ expansion │ │ │ │ cylinders, │ │ │ │ consuming │ │ │ │ an │ 495,000│ 70,710│ 114.4│ 26 equivalent │ │ │ │ to 4 lbs. │ │ │ │ of best │ │ │ │ coal per │ │ │ │ horse-power│ │ │ │ per hour │ │ │ │ ─────────────────┴─────────────┴──────────────┴─────────────┴──────────
TABLE B. VARIOUS ENGINES IN WHICH THE TEMPERATURE OF THE BOILER IS 121° C.[74] AND THAT OF THE CONDENSER 30° C. _Theoretical Duty for each Unit of Heat transmitted, 371 foot-pounds._ ─────────────────┬─────────────┬──────────────┬─────────────┬────────── CASES. │Work produced│Work produced │Work produced│Percentage │for each lb. │ for each lb. │for each unit│ of │ of coal │ of water │ of heat │theoretical │ consumed. │ evaporated. │transmitted. │ duty. ─────────────────┼─────────────┼──────────────┼─────────────┼────────── │ Ft.-lbs. │ Ft.-lbs. │ Ft.-lbs. │ (3) French │ │ │ │ engines, │ │ 98,427│ 160.3│ 43.2 according │ │ │ │ to contract│ │ │ │ (4) English │ │ │ │ engines, │ 565,700│⁶¹⁴⁄₆₁₈×80,814│ 130.8│ 35 according │ │ │ │ to contract│ │ │ │ (6) Common │ │ │ │ engines, │ │ │ │ consuming │ │ │ │ 12 lbs. of │ 165,000│⁶¹⁴⁄₆₁₈×23,571│ 38.1│ 10.3 coal per │ │ │ │ horse-power│ │ │ │ per hour │ │ │ │ (7) Improved │ │ │ │ engines │ │ │ │ with │ │ │ │ expansion │ │ │ │ cylinders, │ │ │ │ consuming │ │ │ │ an │ 495,000│⁶¹⁴⁄₆₁₈×70,710│ 114.4│ 30.7 equivalent │ │ │ │ to 4 lbs. │ │ │ │ best coal │ │ │ │ per │ │ │ │ horse-power│ │ │ │ per hour │ │ │ │ ─────────────────┴─────────────┴──────────────┴─────────────┴──────────
