minimum economical expansion, a maximum efficiency with varying cut-off, as shown by Isherwood, experimentally, for the marine engine a generation ago (1860).* This expansion for maximum efficiency is seen to be substantially the same for all pressures, with the engine here studied, and under the conditions assumed as those of its operation and is very nearly r = 7, thus corresponding accurately with the deduction of Hallauer for the compound engine, as the result of his comparatively limited experimental researches. A singular deduction, such as this, could be given positive proof by such a concurrence of evidence only as could be secured either by a very large number of observations, or by the construction of the surface in which scattered observations could be grouped and made to sustain each other by falling into a smooth area of which the intersections by vertical planes, in whatever directions, give smooth curves. It is only by such constructions that the point of maximum efficiency for both pressure and expansion can be identified with accuracy at all, or a fact like that just pointed out discovered. It is this conclusion that is here intended especially to be emphasized, and the precise value, or even the exact precision of the illustrative example is here of no special importThe value of the method can be now readily recognized and fully appreciated. The result of this particular observation is the determination of the fact that, assuming the premises to accord with current and average fair practice, we may anticipate a gain in economy by increasing steam pressures in similar cases without known limit, and an economy in water-consumption varying about as the reciprocal of the square root of the boiler pressure. It is thus indicated that the deduced ratio of expansion represents a cut-off of maximum efficiency, the cost of operation, in fuel and in steam, being greater with either a larger or a smaller ratio of expansion. It is further seen that, under such conditions of operation, the variation in efficiency with varying pressures is less rapid as this best adjustment is approximated, and that, with varying cut-off, the differences are greater as the pressures are lower, while becoming comparatively small with the highest pressures. A drop of water would trace, by its flow on this model, a line of minima and of best adjustments for the whole series of pressures. High pressures ance. * Experimental Researches in Steam Engineering. are thus less subject to wastes from variable loads with this engine than are low pressures. If we were to apply it to electric street railway work we should adopt the highest pressures practicable. The premises here taken are, in the opinion of the writer, fairly representative of good average practice, and these deductions are probably fairly accurate for the cases taken. Still another, and perhaps even a more important, and certainly more curious and interesting problem may be solved, and has been solved, for this engine by this system of glyptic representation. This problem bears the same relation to the Rankine problem of the best point of cut-off, as a problem in finance, that the problems of maxima and minima in the differential calculus bear to those solved by the application of the calculus of variations. As already seen, the graphic representation of a series of observations by means of a curve, as in the case of the binary alloys and of the curves of water-consumption which have been described, permits the identification of the exact location of a minimum or of a maximum, even though that point may be not even approximate to any computed or observed point identified for the curve. Similarly, the employment of the glyptic system of representation of variations of three quantities, as here illustrated, in the relief-model of the ternary alloys, and in these engine-efficiency surfaces, affords a means of determining accurately the curve of best result in any given case of variation of either pair of variables, and also the maximum or the minimum in that curve, and for the whole surface. It may even happen that there are two or more such curves or such minima or maxima. The final problem, in the present case, is the following: To find the best adjustment of power for the given engine, and the best arrangement of pressures and expansions for that load. The following case is presented as illustrative of the method, not as giving a specific set of values. These should be determined for each case, and by reference to the commercial conditions of that particular case and for the circumstances characteristic of the location and operation of the machine. We proceed thus: Construct a surface which the ordinates, 2, represent the work done per unit weight of water, or unit volume of steam, consumed, the abscissas on the axis of x the cut-off, and those along y the varying steam-pressures. This surface will be found quite simi 66 lar to that already illustrated, but inverted-ordinates here being reciprocal of those-and being convex instead of concave to the base-plane. (Fig. 136.) Let the work performed by the engine at any given pressure and cut-off be represented by the effective horse-power of the machine-its dynamometric power-determined either directly by experiment for a suitable set of observations as to, number and distribution, or by computation of selected cases, giving uniformly distributed ordinates. Horizontal, x 2, planes passed through the surface thus constructed will, by their intersections, give a set of curves of equal power and work, like the profile lines on a topographical relief. The figure here given illustrates such a surface and such lines as obtained for the case in hand. This surface intersects the baseplane at some distance from the origin, its ordinates becoming negative at that line representing those values of the power and work at which the internal and other wastes become so great as to consume all steam supplied, and just turn the engine over at its prescribed speed, without external load at its shaft or on the dynamometer. Vertical planes passed through this surface, parallel to the axis of x, form by their intersections curves of efficiency," as the writer has called them, such as Rankine first described for the ideal case, and such as, for the actual case, the writer has applied to the solution of similar problems relating to the real engine, all wastes being taken into account.* The present problem requires the identification, first of the best of these curves for the given engine and for a stated power, and second the best adjustment of pressure and cut-off for that power and pressure, financial considerations being the controlling conditions. Once this surface is obtained for the proposed engine and for a sufficiently wide range of conditions, the solution of this problem involves simply the examination of the market prices for the locality in which it is to be used, and the questions proposed for solution are quickly and easily answered. In many cases these quantities are not ascertainable with perfect accuracy; but they may almost invariably be determined with sufficient approximation for practical purposes, and even an approximate solution of such problems is vastly better than the usual "guess" of the builder, who invariably seeks maximum duty and usually gives altogether too large an engine for highest commercial efficiency.† * Manual of the Steam Engine. Part i., chap. 7. + Ibidem. |