Chapter 2

carbon, hydrogen, and oxygen being taken at their weight per cent. in the fuel. Strictly speaking, marsh gas should be separately determined. It often happens that available energy is not in a form in which it can be applied directly to our needs. The water flowing down from the mountains in the neighborhood of the Alpine tunnels was competent to provide the power necessary for boring through them, but it was not in a form in which it could be directly applied. The kinetic energy of the water had first to be changed into the potential energy of air under pressure, then, in that form, by suitable mechanism, it was used with signal success to disintegrate and excavate the hard rock of the tunnels. The energy resulting from combustion is also incapable of being directly transformed into useful motive power; it must first be converted into potential force of steam or air at high temperature and pressure, and then applied by means of suitable heat engines to produce the motions we require. It is probably to this circumstance that we must attribute the slowness of the human race to take advantage of the energy of combustion. The history of the steam engine hardly dates back 200 years, a very small fraction of the centuries during which man has existed, even since historic times.

The apparatus by means of which the potential energy of fuel with respect to oxygen is converted into the potential energy of steam, we call a steam boiler; and although it has neither cylinder nor piston, crank nor fly wheel, I claim for it that it is a veritable heat engine, because it transmits the undulations and vibrations caused by the energy of chemical combination in the fuel to the water in the boiler; these motions expend themselves in overcoming the liquid cohesion of the water and imparting to its molecules that vigor of motion which converts them into the molecules of a gas which, impinging on the surfaces which confine it and form the steam space, declare their presence and energy in the shape of pressure and temperature. A steam pumping engine, which furnishes water under high pressure to raise loads by means of hydraulic cranes, is not more truly a heat engine than a simple boiler, for the latter converts the latent energy of fuel into the latent energy of steam, just as the pumping engine converts the latent energy of steam into the latent energy of the pumped-up accumulator or the hoisted weight.

If I am justified in taking this view, then I am justified in applying to my heat engine the general principles laid down in 1824 by Sadi Carnot, namely, that the proportion of work which can be obtained out of any substance working between two temperatures depends entirely and solely upon the difference between the temperatures at the beginning and end of the operation; that is to say, if T be the higher temperature at the beginning, and _t_ the lower temperature at the end of the action, then the maximum possible work to be got out of the substance will be a function of (T-_t_). The greatest range of temperature possible or conceivable is from the absolute temperature of the substance at the commencement of the operation down to absolute zero of temperature, and the fraction of this which can be utilized is the ratio which the range of temperature through which the substance is working bears to the absolute temperature at the commencement of the action. If W = the greatest amount of effect to be expected, T and _t_ the absolute temperatures, and H the total quantity of heat (expressed in foot pounds or in water evaporated, as the case may be) potential in the substance at the higher temperature, T, at the beginning of the operation, then Carnot's law is expressed by the equation:

/ T - t \ W = H( ------- ) \ T /

I will illustrate this important doctrine in the manner which Carnot himself suggested.

Fig. 2 represents a hillside rising from the sea. Some distance up there is a lake, L, fed by streams coming down from a still higher level. Lower down on the slope is a millpond, P, the tail race from which falls into the sea. At the millpond is established a factory, the turbine driving which is supplied with water by a pipe descending from the lake, L. Datum is the mean sea level; the level of the lake is T, and of the millpond _t_. Q is the weight of water falling through the turbine per minute. The mean sea level is the lowest level to which the water can possibly fall; hence its greatest potential energy, that of its position in the lake, = QT = H. The water is working between the absolute levels, T and _t_; hence, according to Carnot, the maximum effect, W, to be expected is--

/ T - t \ W = H( ------- ) \ T / / T - t \ but H = QT [therefore] W = Q T( ------- ) \ T /

W = Q (T - t),

that is to say, the greatest amount of work which can be expected is found by multiplying the weight of water into the clear fall, which is, of course, self-evident.

Now, how can the quantity of work to be got out of a given weight of water be increased without in any way improving the efficiency of the turbine? In two ways:

1. By collecting the water higher up the mountain, and by that means increasing T.

2. By placing the turbine lower down, nearer the sea, and by that means reducing _t_.

Now, the sea level corresponds to the absolute zero of temperature, and the heights T and _t_ to the maximum and minimum temperatures between which the substance is working; therefore similarly, the way to increase the efficiency of a heat engine, such as a boiler, is to raise the temperature of the furnace to the utmost, and reduce the heat of the smoke to the lowest possible point. It should be noted, in addition, that it is immaterial what liquid there may be in the lake; whether water, oil, mercury, or what not, the law will equally apply, and so in a heat engine, the nature of the working substance, provided that it does not change its physical state during a cycle, does not affect the question of efficiency with which the heat being expended is so utilized. To make this matter clearer, and give it a practical bearing, I will give the symbols a numerical value, and for this purpose I will, for the sake of simplicity, suppose that the fuel used is pure carbon, such as coke or charcoal, the heat of combustion of which is 14,544 units, that the specific heat of air, and of the products of combustion at constant pressure, is 0.238, that only sufficient air is passed through the fire to supply the quantity of oxygen theoretically required for the combustion of the carbon, and that the temperature of the air is at 60° Fahrenheit = 520° absolute. The symbol T represents the absolute temperature of the furnace, a value which is easily calculated in the following manner: 1 lb. of carbon requires 2-2/3 lb. of oxygen to convert it into carbonic acid, and this quantity is furnished by 12.2 lb. of air, the result being 13.2 lb. of gases, heated by 14,544 units of heat due to the energy of combustion; therefore:

14,544 units T = 520° + ------------------ = 5,150° absolute. 13.2 lb. X 0.238

The lower temperature, _t_, we may take as that of the feed water, say at 100° or 560° absolute, for by means of artificial draught and sufficiently extending the heating surface, the temperature of the smoke may be reduced to very nearly that of the feed water. Under such circumstances the proportion of heat which can be realized is

5,150° - 560° = --------------- = 0.891; 5,150°

that is to say, under the extremely favorable if not impracticable conditions assumed, there must be a loss of 11 per cent. Next, to give a numerical value to the potential energy, H, to be derived from a pound of carbon, calculating from absolute zero, the specific heat of carbon being 0.25, and absolute temperature of air 520°:

Units. 1 lb. of carbon X 0.25 X 520 = 130 12.2 of air X 0.238 X 520 = 1,485 Heat of combustion = 14,544 ------ 16,159 Deduct heat equivalent to work of \ displacing atmosphere by products of } combustion raised from 60° to 100°, } 32 or from 149.8 cubic feet to 161.3 } cubic feet, / ------ Total units of heat available 16,127

Equal to 16.69 lb. of water evaporated from and at 212°. Hence the greatest possible evaporation from and at 212° from a lb. of carbon--

16,159 u. X 0.891 - 32 u. W = --------------------------- = 14.87 lb. 966 u.

I will now take a definite case, and compare the potential energy of a certain kind of fuel with the results actually obtained. For this purpose the boiler of the eight-horse portable engine, which gained the first prize at the Cardiff show of the Royal Agricultural Society in 1872, will serve very well, because the trials, all the details of which are set forth very fully in vol. ix. of the _Journal_ of the Society, were carried out with great care and skill by Sir Frederick Bramwell and the late Mr. Menelaus; indeed, the only fact left undetermined was the temperature of the furnace, an omission due to the want of a trustworthy pyrometer, a want which has not been satisfied to this day.[2]

[Footnote 2: In the fifty-second volume of the _Proceedings_ (1887-78), page 154, will be found a remarkable experiment on the evaporative power of a vertical boiler with internal circulating pipes. The experiment was conducted by Sir Frederick Bramwell and Dr. Russell, and is remarkable in this respect, that the quantity of air admitted to the fuel, the loss by convection and radiation, and the composition of the smoke were determined. The facts observed were as follows:

Steam pressure 53 lb................................... = 300.6° F. lb. Fuel--Water in coke and wood........................... 26.08 Ash.............................................. 10.53 Hydrogen, oxygen, nitrogen, and sulphur.......... 7.18 ------ Total non-combustible..................... 43.79 Carbon, being useful combustible................. 194.46 ------ Total fuel................................ 238.25

Air per pound of carbon................................ 17-1/8 lb. Time of experiment..................................... 4 h. 12 min. Water evaporated from 60° into steam at 53 lb. pressure 1,620 lb. Heat lost by radiation and convection.................. 70,430 units. Mean temperature of chimney............................ 700° F. " " " air................................ 70° F.

No combustible gas was found in the chimney.

I will apply Carnot's doctrine to this case.

Potential energy of the fuel with respect to absolute zero: Units. 239.25 lb. × 530° abs. × 0.238 ...................... = 30,053 194.46 lb. × 17-1/8 × 530° × 0.238, the weight and heat of air....................... 420,660 194.46 × 14,544 units heat of combustion of carbon... 2,828,200 --------- Total energy 3,278,813 Heat absorbed in evaporating 26.08 lb. of water in fuel............................................ -29,888 --------- Available energy.......................... 3,248,425

Temperature of furnace--

The whole of the fuel was heated up, but the heat absorbed in the evaporation of the water lowered the temperature of the furnace, and must be deducted from the heat of combustion.

Units. Heat of combustion................................... 2,828,200 " " evaporation of 26.08 lb. water............... -29,888 --------- Available heat of combustion.............. 2,798,312

Dividing by 238.25 lb. gives the heat per 1 lb. of fuel used................................... = 11,745 units. And temperature of furnace: 11,745 units/(18.125 lb. × 0.238) + 530°......... = 3,253° Temperature of chimney 700° + 460°............... = 1,160° Maximum duty (3,253° - 1,160°)/3,253°............ = 0.643°

Work of displacing atmosphere by smoke at 700°: Cubic feet. Volumes of gases at 70°........................ = 228.3 " " " " 700°........................ = 499.8 ----- Increase of volume.................... 271.5

Units. Work done= (194.46 lb. × 271.5 cub. ft. × 144 sq. in. × 15 lb.) /722 units ..................................... = 147,720 Maximum amount of work to be expected = 3,248,425 × 0.643.............................. = 2,101,700 Deduct work of displacing atmosphere............. = 147,720 --------- Available work........................ 1,953,980

Actual work done: Units. 1,620 lb. of water raised from 60° and turned into steam at 53 lb..... ...................... = 1,855,900 Loss by radiation and convection................. 70,430 10-1/2 lb. ashes left, say at 500°............... 1,129 --------- Total work actually done.............. 1,927,459 Unaccounted for.................................. 26,521 --------- Calculated available work........................ 1,953,980

The unaccounted-for work, therefore, amounts to only 1½ per cent. of the calculated available work.

Sir Frederick Bramwell ingeniously arranged his data in the form of a balance sheet, and showed 253,979 units unaccounted for; but if from this we deduct the work lost in displacing the air, the unaccounted-for heat falls to less than 4 per cent. of the total heat of combustion. These results show how extremely accurate the observations must have been, and that the loss mainly arises from convection and radiation from the boiler.]

The data necessary for our purpose are:

Steam pressure 80 lb. temperature 324° = 784° absolute. Mean temperature of smoke 389° = 849° " Water evaporated per 1 lb of coal, from and at 212° 11.83 lb. Temperature of the air 60° = 520° absolute. " of feed water 209° = 669° " Heating surface 220 square feet. Grate surface 3.29 feet. Coal burnt per hour 41 lb.

The fuel used was a smokeless Welsh coal, from the Llangennech colleries. It was analyzed by Mr. Snelus, of the Dowlais Ironworks, and in Table II. are exhibited the details of its composition, and the weight and volume of air required for its combustion. The total heat of combustion in 1 lb of water evaporated:

= 15.06 × (0.8497 + 4.265 × (0.426 - 0.035/8)) = 15.24 lb. of water from and at 212° = 14,727 units of heat.

TABLE II.--PROPERTIES OF LLANGENNECH COAL.

+----------+------------+---------------------+ | | | | | | | Products of | | | Oxygen | Combustion at 32° F.| | Analyses | required +--------+------------+ | of 1 lb. | for | | | | of Coal. | Combustion.| Cubic | Volume | | | Pounds. | feet. | per cent. | ---------------------+----------+------------+--------+------------+ Carbon........... | 0.8497 | 2.266 | 25.3 | 11.1 | Hydrogen......... | 0.0426 | 0.309 | 7.6 | 3.4 | Oxygen........... | 0.0350 | --- | --- | --- | Sulphur.......... | 0.0042 | --- | --- | _ --- | Nitrogen......... | 0.1045 | --- | 0.18 | | | Ash.............. | 0.0540 | --- | --- | | | +----------+------------+ | | 85.5 | | | | | | | Total........... | 1.0000 | 2.572 | --- | | | 9-1/3.lb nitrogen | --- | --- | 118.9 | | | 6 lb. excess of air. | --- | --- | 71.4 | _| | +----------+------------+--------+------------+ Total cubic feet of | | | | | products per 1 lb. | | | | | of coal........... | -- | -- | 226.4 | 100.0 | ---------------------+----------+------------+--------+------------+

The temperature of the furnace not having been determined, we must calculate it on the supposition, which will be justified later on, that 50 per cent more air was admitted than was theoretically necessary to supply the oxygen required for perfect combustion. This would make 18 lb. of air per 1 lb. of coal; consequently 19 lb. of gases would be heated by 14,727 units of heat. Hence:

14,727 u. T = ---------------- = 3,257° 19 lb. × 0.238

above the temperatures of the air, or 3,777° absolute. The temperature of the smoke, _t_, was 849° absolute; hence the maximum duty would be

3,777° - 849° --------------- = 0.7752. 3,777°

The specific heat of coal is very nearly that of gases at constant pressure, and may, without sensible error, be taken as such. The potential energy of 1 lb. of coal, therefore, with reference to the oxygen with which it will combine, and calculated from absolute zero, is:

Units. 19 lb. of coal and air at the temperature of the air contained 19 lb. × 520° × 0.238 2,350 Heat of combustion 14,727 ------- 17,078 Deduct heat expended in displacing atmosphere 151 cubic feet - 422 ------ Total potential energy 16,656

Hence work to be expected from the boiler:

/ 3,777° - 849° \ = 17,078 units X ( --------------- ) - 422 units \ 3,777° / ---------------------------------------------- = 13.27 lb. 966 units

of water evaporated from and at 212°, corresponding to 12,819 units. The actual result obtained was 11.83 lb.; hence the efficiency of this boiler was

11.83 ------- = 0.892. 13.27

I have already claimed for a boiler that it is a veritable heat engine, and I have ventured to construct an indicator diagram to illustrate its working. The rate of transfer of heat from the furnace to the water in the boiler, at any given point, is some way proportional to the difference of temperature, and the quantity of heat in the gases is proportional to their temperatures. Draw a base line representing -460° Fahr., the absolute zero of temperature. At one end erect an ordinate, upon which set off T = 3,777°, the temperature of the furnace. At 849° = _t_, on the scale of temperature, draw a line parallel to the base, and mark on it a length proportional to the heating surface of the boiler; join T by a diagonal with the extremity of this line, and drop a perpendicular on to the zero line. The temperature of the water in the boiler being uniform, the ordinates bounded by the sloping line, and by the line, _t_, will at any point be approximately proportional to the rate of transmission of heat, and the shaded area above _t_ will be proportional to the quantity of heat imparted to the water. Join T by another diagonal with extremity of the heating surface on the zero line, then the larger triangle, standing on the zero line, will represent the whole of the heat of combustion, and the ratio of the two triangles will be as the lengths of their respective bases, that is, as (T - _t_) / T, which is the expression we have already used. The heating surface was 220 square feet, and it was competent to transmit the energy developed by 41 lb. of coal consumed per hour = 12,819 u. × 41 u. = 525,572 units, equal to an average of 2,389 units per square foot per hour; this value will correspond to the mean pressure in an ordinary diagram, for it is a measure of the energy with which molecular motion is transferred from the heated gases to the boiler-plate, and so to the water. The mean rate of transmission, multiplied by the area of heating surface, gives the area of the shaded portion of the figure, which is the total work which should have been done, that is to say, the work of evaporating 544 lb. of water per hour. The actual work done, however, was only 485 lb. To give the speculations we have indulged in a practical turn, it will be necessary to examine in detail the terms of Carnot's formula. Carnot labored under great disadvantages. He adhered to the emission theory of heat; he was unacquainted with its dynamic equivalent; he did not know the reason of the difference between the specific heat of air at constant pressure and at constant volume, the idea of an absolute zero of temperature had not been broached; but the genius of the man, while it made him lament the want of knowledge which he felt must be attainable, also enabled him to penetrate the gloom by which he was surrounded, and enunciate propositions respecting the theory of heat engines, which the knowledge we now possess enables us to admit as true. His propositions are:

1. The motive power of heat is independent of the agents employed to develop it, and its quantity is determined solely by the temperature of the bodies between which the final transfer of caloric takes place.

2. The temperature of the agent must in the first instance be raised to the highest degree possible in order to obtain a great fall of caloric, and as a consequence a large production of motive power.

3. For the same reason the cooling of the agent must be carried to as low a degree as possible.

4. Matters must be so arranged that the passage of the elastic agent from the higher to the lower temperature must be due to an increase of volume, that is to say, the cooling of the agent must be caused by its rarefaction.