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Piston speed, as usually reckoned-the length of stroke in feet multiplied by the number of strokes per minute, giving the velocity of piston in feet per minute—is a mean velocity, agreeing with the actual velocity only at two instants during each stroke. Multiplied into the uniform pressure acting on the area of the piston, it gives the power correctly; but multiplied into the mean pressure, obtained by integrating the widely-varying pressures of a diagram, with a high ratio of expansion, it does not of necessity give the power correctly.

A familiar example of the fallacy of averaging two sets of co-ordinates is to be found in the different areas of the two sides of a piston. The area of cross-section of piston rod reduces the area of the side nearest to the crank usually almost three per cent. It is customary in crude calculations to take the mean area-that is, to deduct half the area of piston rod cross-section from the area of crosssection of the cylinder, and this gives a correct result in the rare instances when the mean effective pressure is equal on the two sides; in all other cases it introduces an amount of error proportioned to the inequality of mean effective pressure-an error often quite important in estimating the performance of an engine. Now the variations of pressure, with high ratios of expansion, are very great, and it cannot be a matter of indifference that the pressure acting on the piston half the time, in the end quadrants, at piston speed below the mean, is much above the mean pressure: while the pressure acting on the piston the other half of the time in the mid stroke quadrants, at piston speed above the mean, is below the mean pressure.

Nor does the beautiful and important action of inertia, in the reciprocating parts, altogether remedy this inequality of pressure. Between the two end quadrants, or half quadrants, indeed, it may be made to produce substantial equality, and between the mid-stroke quadrant and the half quadrant at the ends, the inequalities may so be greatly reduced, but only by diminishing the pressure on the crank at one end, and correspondingly increasing it at the other end. Between the two ends of the mid-stroke quadrant, too, the pressure may be completely equalized by this agency. But nothing can be transferred from the high initial pressure in the first half quadrant, to the lower pressure at mid-stroke, produced by high ratios of expansion. This figure, 18, like the last, of which, indeed, it is only an exemplification, can lay no claim to geometrical accuracy, and leads to no trustworthy numerical results. The tables which follow have, therefore, little value, but are inserted as only another manner of presenting the substance of this and the preceding diagram. The results are the ratios of areas by planimeter

measurement:

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Fig. 19. This represents an ideal diagram with twelve-fold expansion; initial pressure, P, 120 lbs. + atm.=134.7 absolute; terminal pressure, P=11.225 lbs. abs.; back pressure P=2.7 lbs. absolute.

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The horizontal pressure on the piston is represented by the hyperbolic curve P12, A", P, 1; the horizontal pressure on the crank pin, by the line P', AA", P', and the tangential pressure on the crank pin by the line S A' A" S'.

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The horizontal pressure on the crank pin is exactly the same as that on the piston (no account being here taken of friction), as the pressure is distributed upon the several parts of the stroke, as much being added to the last half of the diagram of piston pressure as is taken from the first half, upon the assumption that the inertia of reciprocating parts at dead center is equal to one half the absolute initial pressure=67.35 lbs. per square inch of piston area. The rotative effect is nearly, perhaps exactly, the same upon the crank pin, whether reckoned from the horizontal diagram of piston or crank pressure.

The slight difference of 2.5 per cent., which appears in the subjoined table (25.93 and 26.27), may result from some slight error in my numerical calculations, which I have not been able to detect. see no reason why they should not be alike.

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The pressures in the foregoing table are all absolute. I assume that back pressure on the piston, occasioned by pressure in the condenser, will be 2.7 pounds. This must be subtracted from all horizontal pressures, and a due proportion of it from all rotative pressures, and the result will be seen in the next table:

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It will not be overlooked that the subtraction of a constant from variable pressures, changes materially their mutual proportions. For example, the rotative effect, last 0.15 of stroke, unmodified diagram, (fourth column, fourth line), is reduced no less than twenty-two per cent., while that in the first 0.15 of stroke (fourth column, first line), is reduced only 2.2 per cent., only one-tenth as much in proportion to its magnitude.

It will be seen by a comparison of the four first numbers in the last column of the foregoing tables, that the rotative effect of the first and last quadrants is pretty nearly equalized, but that a little too much has been transferred from the first part of the stroke to the last. In the mid stroke, considerably too much has been transferred from the first to the last half. Of course, the remedy is to give less value to the inertia of the reciprocating parts. The weight of these parts can only be varied within very narrow limits, and will generally range between 2.5 and 4.0 pounds per square inch of cylinder cross section. But the speed is usually determined by the judgment of the engineer or designer of the engine, and should be carefully adjusted to the requirements of the diagram.

Clearance and compression have been omitted, to avoid needless complication.

Of course, both have their importance, but their introduction here would, it is thought, only tend to confuse.

It may not be out of place here to call attention to the relation of the area representing rotative effect, to the diagram of horizontal piston pressure upon which it is traced.

Referring to Figs. I-V, it was correctly stated that the ratio of the area of rotative effect is to the area of the circumscribing parallelogram, square, lozenge, or other figure representing horizontal

π

piston pressure, as to unity. Yet this quantity, say 0.7854, (radius

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or crank being unity) if multiplied by the mean piston pressure and area, and by the velocity of the crank, will give a result, as power, considerably in excess of the true result, to be obtained by multiplying the effective piston pressure, area and mean velocity continuously together. In fact, the true coefficient of rotative effect is π =0.63662, instead of 0.7854.

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π

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The reason for this is, that the area is to be obtained by integrating the sines of angles corresponding to equal successive differences of co-sines (ds), while the mean rotative pressure is to be obtained by integrating the sines corresponding to equal successive differences of arc (da).

Fig. 25 illustrates this. The arithmetical mean of the numbers representing the length of sines at .1, .2, .3, etc., dividing OS' into

π

ten equal parts, will give 0.7761, which differs but little from and

4

would agree with it exactly if taken at infinitesimal intervals. On the other hand, the corresponding mean of the sines of the equal angles, 9°, 18°, 27°, etc., which divide the arc SC into ten equal parts, will give 0.6352, a still closer approximation to ,with which

2

π

it would exactly coincide if the divisions of the arc were sufficiently minute.

It follows, then, that while the curves of rotative effect correctly represent this force upon the diagram, its mean value, for multiplying into piston area and crank speed to obtain the power, must be found by multiplying the mean effective pressure on the piston by 0.63662: or by dividing it by the reciprocal of that number π =1.5708-the ratio which crank speed bears to piston speed.

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We may now make a more particular application of the foregoing principles to a comparison of the distribution of the unequal pressures, resulting from high ratios of expansion, in single and in compound engines.

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Fig. 19. This figure will require but little explanation. It represents a diagram with twelve-fold expansion, from 134.7 pounds absolute at P1, to 11.225 pounds at P2 in a single cylinder. The 12, horizontal pressure P',AA" P' and the tangential pressure SA' A'S', are determined upon the assumption that the weight and velocity of the reciprocating parts are such as to absorb half the absolute initial pressure on the piston, and transfer it to the end of the stroke.

This, it will be seen, shows much more than half the work done in the last half stroke; but such excess might be reduced to any desired extent by reducing the speed.

The double vertical lines marked 45° and 145° about fifteen per cent. from each end, divide the mid-stroke quadrant from the two end half quadrants.

Fig. 20. The same initial pressure, ratio of expansion, terminal pressure, speed, effect of reciprocating parts, and rotative effect that we saw in the last figure carried out in a single cylinder, are here shown as operating in a compound engine, in which the cylinder volumes are to each other in the ratio of one to four. Except the effect of clearances, compression, early release, expansion between cylinders, and, perhaps, other minor causes of irregularity or loss, not necessary to consider here, the diagram of the low-pressure cylinder fills the space occupied by back pressure in the high-pressure cylinder, and the two diagrams produce, with twelve-fold expansion, a combined diagram with three-fold expansion. Acting with a net increase of effect three-fold, by reason of the net increase of volume, it will come to the same thing, if we consider the stroke and area of piston equal, and the mean effective pressure in the lowpressure cylinder three times as great as it actually is. In the highpressure cylinder, initial pressure being 134.7 pounds, expansion three-fold, and terminal pressure 44.9 pounds, the mean absolute pressure is 94.25 pounds. In the low-pressure cylinder, the initial pressure is 44.9 pounds, the ratio of expansion four, terminal pressure, 11,225 pounds, and mean total pressure, 20.74. As this latter is the back pressure in the high-pressure cylinder, the mean effective pressure in that cylinder is 94.25-20.74-73.51. Now,

73.51+(3x20.74)
94.25

=1.44, so that, with constant stroke,

if we take the area of what we may call the equivalent cylinder, 1.44 times that of the high-pressure cylinder, we may consider the whole expansion precisely as if it all took place in such equivalent cylinder, with only three-fold expansion. Of course, taking back pressure in low-pressure cylinder, clearance, compression, etc., into con

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