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had a positive action. A similar series of experiments was made for each of the pressures referred to.

The general shape of curves showing the results of experiments may be obtained by plotting the detailed results on a sheet of cross section paper. The points will not exactly fall in a curve in all cases, as some variations will occur in all experimental work which can only be eliminated by repeating the experiments at points where an apparent irregularity in the curve creates doubt. When, as in this case, the experiments plot in curves having similar general features, the problem is simplified by the fact that one curve may be corrected to a certain extent at doubtful points from another. In this particular case a typical curve D, for instance, was compared with calculated curves G and H, No. 2, based on the positive requirements of the problem, which could not be changed by mechanism of any kind, thus again enabling the lines to be located between the various points representing the experimental data so as to reduce the probable errors for each case to a very small limit. The points upon which the curves were founded are not produced here, as they would only confuse the diagram. It may be stated that the curves for 40, 60 and 80 pounds pressure were satisfactorily located throughout. The curve for 25 pounds pressure showed greater variations, but it was easily brought into satisfactory position by comparison with the others. The curve for 100 pounds could not, from lack of resistance and boiler power, be developed for the very lowest degrees of expansion, but with the check of the other curves no further experiments could have changed its shape or position materially.

Curves No. 2 show that the quantity of steam actually required. in excess of the calculated quantity is very nearly the same at all points of cut-off, though increasing somewhat as the expansion is increased. As, however, the weight of steam required per horse power decreases as the expansion increases, up to the point of inflection, the quantities of steam required in excess of the calculated quantities are much larger percentages of the total costs with the higher degrees of expansion. In this connection it will be interesting to note that corresponding series of experiments were made in the same engine with a vacuum, in which case the quantities of steam in excess of the calculated quantities were very much in excess of those shown without a vacuum, corresponding to the conditions in non-condensing engines. In fact, the losses were so great that no economy resulted from the vacuum for steam

pressures of 80 pounds or upwards when using an engine as small as that experimented with.

In order to show a practical application of the experimental curves there are represented in the space designated "No. 3," the approximate forms of two indicator diagrams with the same mean pressure, the one designated K showing an initial steam pressure of 80 pounds cut off at one-quarter of the stroke, the other designated J being the typical form of diagram resulting from the use of the throttle valve with main valve cutting off at of the stroke. The cost of the power may be found by referring to the small crosses a and b in series No. 1, from which it will appear that the cost for 80 pounds initial pressure cut off at of the stroke would be 35 pounds of water per horse power per hour as shown at, and for a steam pressure throughout the stroke equal to the initial pressure of 53 pounds on diagram J, No. 3, cut off at of the stroke, the cost, as shown at a on diagram No. 1, would be nearly 56 pounds of water per horse power per hour, but in practice this would be reduced somewhat on account of the expansion obtained by the wire drawing or throttling shown by the inclination of the upper line of diagram J, as will be explained hereafter. The comparison shows the advantage of using a cut-off to reduce the power of an engine of given size rather than the throttle. To secure the best possible economy with the throttle the main valve should be provided with sufficient lap to cut off the steam at of the stroke or less, and the space in the steam chest intervening between the main valve and the throttle valve be reduced as much as possible. In this way the chest will be filled to boiler pressure during the period of cut-off, and though this will be reduced as soon as the main valve opens at the beginning of the stroke there will be secured a certain degree of expansion due to wire drawing which will reduce the cost proportionally to the actual degree of expansion secured. To determine the probable cost under these circumstances we may assume that the results which would be obtained with a diagram like will be the same as if steam at the actual initial pressure shown were expanded a sufficient number of times to produce the actual terminal pressure. In this particular case there would be an equivalent expansion of 1.42 times, so the probable cost would be reduced to 52.0 pounds of water per horse power per hour, and similarly the results due to a greater reduction of pressure by wire drawing may be ascertained. In

many instances plain slide valve engines are used which are too large for the work. In such case the mean pressures are low and the costs of the power very great compared with what they would be in engines of proper size using steam expansively. For instance, in the engine with which the experiments were made the cost of the power with 80. pounds of steam cut off at of the stroke was only 35 pounds of feed water; but when the engine was operated with an initial pressure of 40 pounds, with a relative expansion of 2, the cost as shown was 52 pounds of feed water, and if steam had been cut off relatively at of the stroke the cost would have been over 60 pounds of feed water. It is not uncommon, however, to have still lower pressures in the steam chest which would run the cost up to 70 pounds of feed water as shown by the curve A. The illustrations indicate how many comparisous of interest and value may be made by the inspection of curves showing only the results with a small engine.

The tables referred to may be consulted for illustrations covering a little broader field, but are insufficient to embrace the whole subject, and have suggested to the writer the desirability of further analyzing the experiments to ascertain the probable cost. under all customary conditions, and of formulating the results so that the cost for any particular case may be obtained approximately by a short calculation, without consulting bulky tables which, after all, may not contain the desired information for the particular condition under consideration.

It has been thought that the methods necessarily adopted in bringing together in one formula not simply the results shown by a single graphic curve, but those shown by a series of curves, vary themselves by a law necessarily represented by another curve, and their joint variations varying again by a law shown by the ordinates of still another curve, etc., etc., would be of interest to many here present.

The general principle involved in a work of this kind is to first find the equation of a curve representing each of the particular conditions, then to combine the equations by addition, multiplication or substitution, according to the conditions, so as to obtain from one equation the result due to all the conditions. The treatment in this case will be rather from the practical side, using equations of as simple form as possible, though they may not in all cases suit a particular set of conditions as accurately as a curve with more elaborate formula, but the errors thus intro

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