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der; then the best ratio of expansion for the series, all things considered, this study being made from the financial standpoint, as must be every problem which the engineer is called upon to solve. It is not the thermo-dynamic, nor the fluid, nor even the engine efficiency, which must be finally allowed to fix the best ratio of expansion; but it must be the ratio of expansion at maximum commercial efficiency, that which will make the cost of operation at the desired power a minimum for the life of the system. The total ratio being settled upon, and that allowable as a maximum, for a single cylinder, it is at once easy to determine the best number of cylinders in series. The first-mentioned ratio is that at maximum commercial efficiency, as just stated; but the second must be taken as that which gives the highest efficiency of engine, the back pressure in that cylinder, and the friction of the cylinder taken singly, being considered, together with its proper proportion of the friction of the engine as a whole.

Studying the method of distribution of waste among the several cylinders of the multi-cylinder engine, it will be observed that, since the pressures increase more rapidly than the temperatures, the range of temperature in the high pressure cylinder is greatest; while, the same weight of steam passing through the whole series, the low-pressure cylinder presents the

largest area of condensing surface in proportion to steam used. These differences are to a certain extent, though not wholly, compensatory. It may be assumed, however, without serious error, that the necessity of applying jackets or other methods of reducing internal wastes, will apply substantially as imperatively to one cylinder as to another; and that the adoption of a common ratio of expansion for both or all cylinders, or of apportioning the rates with reference to the equal division of power among them, will be found perfectly admissible, and will introduce no serious avoidable loss. Authorities have greatly differed in their views as to the relative advantage of jacketing one or another cylinder; but it is at least safe to jacket all, and probably, as above indicated, best to do so. The importance of the jacket evidently becomes less as other expedients for reducing wastes of this kind are adopted and are made more effective; as by increasing speed of engine, by superheating, by reheating between cylinders; and cases may be imagined in which the jackets may cease to have sufficient value to justify the acceptance of the risks and expense incurred in their employment. The same is true of the more complicated forms of valve gear needed to secure an approximation to the ideal distribution of steam,

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CHAPTER II.

EXPANSION OF STEAM.

IT is well known that the amount of work which can be obtained from a given weight of steam, when not used expansively, is practically independent of the pressure at which the steam is worked. Thus, if steam of 100 pounds per square inch absolute pressure be admitted behind a piston one square foot in area, then by the time the piston has been pushed through 4.38 feet, exactly one pound of steam has been admitted, and the work done by it during admission is equal to the pressure on the piston multiplied by the number of feet through which the piston moves, or 100 x 144 × 4.38 = 63,072 foot pounds. If, then, the steam escapes into the air, this represents the total work done by the steam.

Suppose now, that the steam used had been at a pressure of twenty pounds per square inch, and that this steam had been admitted behind a piston one square foot in area, then by the time the piston had been pushed through 19.7 feet, exactly one pound of steam at twenty pounds absolute measure has been admitted, and the work done by it during admission is equal to

20 × 144 × 19.7 = 56,736 foot pounds. But the pound of steam at 100 pounds pressure did 63,072 foot pounds, which is not much greater than that done by the pound of steam at twenty pounds pressure. No advantage, however, has here been taken of the expansive power of steam, which increases as its initial pressure increases. Suppose we wish the terminal pressure in the cylinder to be twenty pounds absolute, then evidently the 1 pound of steam at 20 pounds cannot do any more work on the piston than is given above, namely, 56,736 foot pounds. On the other hand, the pound of steam at 100 pounds pressure can do a great deal more work, if, instead of exhausting into the air at that pressure, it is allowed to push the piston forward till it expands and becomes reduced in pressure to twenty pounds. This capacity for doing more work must have struck the most careless observer when standing by the exhaust pipe of a high-pressure engine.

Now the steam at 100 pounds pressure pushed the piston through 4.38 feet and escaped into the air, but it would push the piston through five times this distance in expanding to a terminal pressure of 20 pounds, or 4.38 × 5 = 21.90 feet, which is nearly the same distance as the I pound of steam at 20 pounds pushed the piston, namely, 19.7 feet. The piston in the two cases has been pushed through about the same dis

tance by the same weight of steam; but observe the difference in the amount of work done in the two cases. In the first case the mean pressure on the piston varies from 100 pounds per square inch at one end of the stroke to 20 pounds per square inch at the other end, giving a mean pressure throughout the stroke of 52.2 pounds. In the second case the pressure on the piston was 20 pounds throughout. Hence the gain by using the high-pressure steam is 52.2÷ 20=2.61; in other words, 2.61 times the amount of work is done by using the high-pressure steam expansively. The significance of this result becomes more apparent when we remember that the I pound of steam at 100 pounds pressure only costs an appreciably small quantity of fuel more for its production than the I pound of steam at 20 pounds so small, in fact, that it may be neglected in practice.

To take another view of the case: Suppose the steam at 100 pounds pressure, instead of having been cut off at one-fifth of the stroke, and expanded to twenty pounds pressure as before, had been supplied to the cylinder at 100 pounds pressure throughout the whole stroke, then the mean pressure would be 100 pounds per square inch, and amount of work done would be five times as great as when steam at 20 pounds was admitted throughout the whole strol-e; but it would only be 52.61, or say twice as great as

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