Part 22

Hence the total quantity added is equal to

Ατ + (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 Βτ, 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.”

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; 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.

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.

(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; 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, 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.

(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.

(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 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 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. 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 │ │ │ │ ─────────────────┴─────────────┴──────────────┴─────────────┴──────────

APPENDIX A. EXTRACTS FROM UNPUBLISHED WRITINGS OF CARNOT.

I. NOTES.